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Normal subgroup

In abstract algebra, a normal subgroup of a group G is a subgroup H that remains unchanged under conjugation by any element of G, satisfying gHg^{-1} = H for all g \in G.^[1] This invariance, also known as being invariant or self-conjugate, distinguishes normal subgroups from ordinary subgroups and is denoted by H \triangleleft G.^[2] An equivalent characterization is that the left and right cosets of H in G coincide, meaning gH = Hg for every g \in G.^[2] In abelian groups, every subgroup is normal due to the commutativity of the group operation.^[1] For non-abelian groups, such as the symmetric group S_3, the alternating subgroup A_3 (consisting of even permutations) serves as a classic example of a normal subgroup.^[2] Normal subgroups are fundamental to group theory because they allow the formation of quotient groups or factor groups, where the cosets of H in G form a new group under the induced operation, providing insight into the structure of G.^[2] This construction is essential for concepts like group homomorphisms, where the kernel of a homomorphism is always normal, and for analyzing solvability and composition series in finite groups.^[1]

Definitions

Formal Definition

A subgroup N of a group G is called normal, denoted N \triangleleft G, if for every g \in G and every n \in N, the conjugate g n g^{-1} also belongs to N. Equivalently, the set g N g^{-1} = \{ g n g^{-1} \mid n \in N \} equals N for all g \in G.^[3] This conjugation condition implies that G acts on N by the map n \mapsto g n g^{-1} for each fixed g \in G, and normality ensures that N is invariant as a set under this group action.^[3] The concept of a normal subgroup was introduced by Évariste Galois around 1832, initially termed an "invariant" subgroup, in his work on the solvability of polynomial equations by radicals; the modern terminology "normal" appeared later, formalized in treatments by mathematicians such as Camille Jordan in 1870.^[3]^[4] To verify normality directly, one checks the conjugation condition for all g \in G and n \in N, often by examining generators of N if finitely generated. In particular, the trivial subgroup \{ e \} (where e is the identity) is always normal, since g e g^{-1} = e \in \{ e \} for all g \in G. Similarly, G itself is normal in G, as conjugation by any g \in G is an automorphism of G, so g G g^{-1} = G.^[3]^[4]

Equivalent Conditions

A subgroup N of a group G is normal if and only if it satisfies the conjugation invariance condition: gNg^{-1} = N for all g \in G. This is equivalent to gng^{-1} \in N for all g \in G and n \in N.^[5]^[6] One standard equivalent condition is that the left and right cosets of N coincide: gN = Ng for all g \in G. To see this, assume gNg^{-1} = N; then multiplying on the right by g gives gN = Ng. Conversely, if gN = Ng, then for any n \in N, gn = ng' for some g' \in G, so gn = ng' = n(gg^{-1})g' = n h g where h = gg^{-1} \in G, but more directly, g n g^{-1} = (g n) g^{-1} \in N g^{-1} = g^{-1} g N g^{-1} = g N g^{-1}, and since g N = N g, it follows that g n g^{-1} \in N. Thus, the coset condition implies conjugation invariance.^[5]^[7] Another equivalent condition is that N is the kernel of some group homomorphism \phi: G \to K for a group K. Kernels are always normal subgroups because if n \in N = \ker \phi, then for any g \in G, \phi(g n g^{-1}) = \phi(g) \phi(n) \phi(g)^{-1} = \phi(g) e \phi(g)^{-1} = e, so g n g^{-1} \in \ker \phi = N. Conversely, if N is normal, the quotient map to G/N (detailed in the Quotient Groups section) has N as its kernel.^[8]^[9] Normality is also equivalent to N containing all commutators of the form [g, n] = g n g^{-1} n^{-1} for g \in G and n \in N, i.e., [G, N] \leq N. To derive this, note that if N is normal, then g n g^{-1} \in N, so [g, n] = (g n g^{-1}) n^{-1} \in N. Conversely, if [G, N] \leq N, then for n \in N, g n g^{-1} = [g, n] n \in N, since both factors are in N.^[10] These conditions are mutually equivalent through the conjugation invariance. For instance, the coset condition implies the commutator condition via g n = n' g for n' = g n g^{-1} \in N, yielding [g, n] = n' n^{-1} \in N. The kernel condition follows from normality enabling the quotient homomorphism.^[5]^[7] A subgroup N is normal if and only if it is a union of conjugacy classes of G. This holds because conjugacy classes are the orbits under conjugation, and normality means N is invariant under conjugation, hence a disjoint union of such orbits (including the identity class). The converse follows directly from the conjugation condition.^[11]^[12]

Examples

In Abelian and Nilpotent Groups

In abelian groups, every subgroup is normal. This property arises from the commutativity of the group operation: for any elements g \in G and n \in N where N is a subgroup, the conjugate g n g^{-1} = n, since g n = n g.^[7] A representative example is the additive group of integers \mathbb{Z}, whose subgroups are of the form n\mathbb{Z} for integers n \geq 0; each such subgroup satisfies the normality condition due to the abelian structure.^[7] Cyclic groups provide further illustration, as they are abelian and their subgroups correspond directly to divisors of the group order. For a cyclic group G of order m, the subgroups are \langle g^{m/d} \rangle for each divisor d of m, and all are normal.^[13] The Klein four-group V_4 \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}, an abelian group of order 4, has three proper nontrivial subgroups, each isomorphic to \mathbb{Z}/2\mathbb{Z}, and all are normal.^[13] To verify normality in an abelian group like \mathbb{Z} \times \mathbb{Z}, consider the subgroup H generated by (2,0), which consists of elements (2k, 0) for k \in \mathbb{Z}. For any (a,b) \in \mathbb{Z} \times \mathbb{Z} and (2k,0) \in H, the conjugate is (a,b) + (2k,0) - (a,b) = (2k,0), confirming H is normal.^[14] Nilpotent groups extend this notion beyond strict abelian cases, featuring characteristic normal subgroups in their upper central series. The upper central series of a group G is the sequence Z_0(G) = \{ e \} \subseteq Z_1(G) \subseteq Z_2(G) \subseteq \cdots, where Z_{k+1}(G)/Z_k(G) is contained in the center of G/Z_k(G), and each Z_k(G) is normal in G.^[15] The quaternion group Q_8 = \{ \pm 1, \pm i, \pm j, \pm k \} with relations i^2 = j^2 = k^2 = ijk = -1 is nilpotent of class 2; its center Z(Q_8) = \{ \pm 1 \} and derived subgroup Q_8' = \{ \pm 1 \} are both normal, as are all its proper subgroups.^[16] In contrast to these structures, non-abelian groups generally possess subgroups that are not normal.

In Symmetric and Alternating Groups

In the symmetric group S_n, the alternating group A_n, consisting of all even permutations, forms a normal subgroup of index 2 for n \geq 3. This normality follows from A_n being the kernel of the sign homomorphism \operatorname{sgn}: S_n \to \{\pm 1\}, which maps each permutation to the parity of its number of inversions (or equivalently, the sign of the permutation); kernels of homomorphisms are always normal subgroups.^[17] Alternatively, since subgroups of index 2 are normal, and |S_n : A_n| = 2 with |A_n| = n!/2, this confirms the result; moreover, A_n is the unique such subgroup because any other index-2 subgroup would also be the kernel of a homomorphism to \mathbb{Z}/2\mathbb{Z}, but the sign homomorphism is the only nontrivial one up to isomorphism.^[18] The trivial subgroup \{[e](/page/E!)\} and S_n itself are always normal in S_n. For n \geq 5, these, along with A_n, are the only normal subgroups of S_n; there are no other proper nontrivial normal subgroups.^[19] In the alternating group A_n, the structure of normal subgroups varies with n. For n \geq 5, A_n is simple, meaning it has no nontrivial proper normal subgroups beyond \{[e](/page/E!)\} and A_n itself. However, for smaller n, exceptions arise: in A_4, the Klein four-group V = \{e, (12)(34), (13)(24), (14)(23)\} is a normal subgroup of order 4, consisting of the identity and the three double transpositions, which form a single conjugacy class in A_4.^[20]^[21] Dihedral groups D_m, which describe the symmetries of a regular m-gon and embed as subgroups of S_m via the action on vertices, provide further examples of normal subgroups in permutation groups. The rotation subgroup \langle r \rangle, generated by a rotation r of order m, has index 2 in D_m and is thus normal. For the specific case of D_4 (order 8, symmetries of the square, embedded in S_4), explicit conjugation verifies this: label vertices 1,2,3,4 clockwise, with r = (1234) and reflections like s = (24); then for any reflection s' (e.g., s' = (13)), s' r s'^{-1} = r^{-1} = (1432) \in \langle r \rangle, confirming closure under conjugation.^[22] A contrasting example occurs in S_3, the symmetric group on 3 letters (order 6, isomorphic to D_3). The subgroup A_3 = \langle (123) \rangle, generated by the 3-cycle and consisting of even permutations \{e, (123), (132)\}, is normal as it coincides with the alternating group of index 2. However, the subgroup generated by a transposition, such as \langle (12) \rangle = \{e, (12)\}, is not normal: conjugation by (13) yields (13)(12)(13)^{-1} = (23) \notin \langle (12) \rangle.^[23]

Properties

Closure and Basic Properties

Normal subgroups exhibit closure properties under basic set operations within the group. Specifically, the intersection of two normal subgroups of a group G is itself a normal subgroup of G. Let N and M be normal subgroups of G. For any g \in G and x \in N \cap M, since N \trianglelefteq G and M \trianglelefteq G, it follows that g^{-1} x g \in N and g^{-1} x g \in M, so g^{-1} x g \in N \cap M. Thus, N \cap M \trianglelefteq G.^[14] The product of two normal subgroups is also normal. Let N, M \trianglelefteq G, and define NM = \{ nm \mid n \in N, m \in M \}. For any g \in G and x = nm \in NM, compute g^{-1} x g = g^{-1} (nm) g = (g^{-1} n g)(g^{-1} m g). Since N \trianglelefteq G and M \trianglelefteq G, g^{-1} n g \in N and g^{-1} m g \in M, so g^{-1} x g \in NM. Therefore, NM \trianglelefteq G. If G is finite, the order of NM satisfies |NM| = |N| |M| / |N \cap M|, as the map N \times M \to NM given by (n, m) \mapsto nm has kernel \{(n, m) \mid nm = e\} \cong N \cap M.^[24]^[25] A normal subgroup consists of entire conjugacy classes. If N \trianglelefteq G and n \in N, then for any g \in G, the conjugate g n g^{-1} \in N by the definition of normality, so the conjugacy class of n is contained in N. Since this holds for every n \in N and N contains the identity (its own conjugacy class), N is a union of conjugacy classes of G.^[26] The index of a normal subgroup relates directly to the group's order via Lagrange's theorem. If N \trianglelefteq G and G is finite, the quotient group G/N has order [G : N], so |G| = |N| \cdot [G : N] and thus [G : N] divides |G|.^[27] Associated with any subgroup are the normal core and normal closure in G. The normal core of a subgroup H \leq G is the largest normal subgroup of G contained in H, given by \mathrm{core}_G(H) = \bigcap_{g \in G} g H g^{-1}. The normal closure of H is the smallest normal subgroup of G containing H.^[28]^[29] Normal subgroups are preserved under group automorphisms in the sense that their images remain normal. If N \trianglelefteq G and \phi \in \mathrm{Aut}(G), then \phi(N) \trianglelefteq G, because automorphisms preserve the group operation and conjugation: for g \in G and n \in N, \phi(g)^{-1} \phi(n) \phi(g) = \phi(g^{-1} n g) \in \phi(N) since g^{-1} n g \in N.^[30]

Lattice of Normal Subgroups

The set of all normal subgroups of a group G, ordered by inclusion, forms a lattice known as the normal subgroup lattice of G.^[31] In this lattice, the meet of two normal subgroups H and K is their intersection H \cap K, which is itself normal in G.^[31] The join of H and K is the subgroup generated by their union, denoted \langle H \cup K \rangle or equivalently HK since both are normal, and this join is also normal in G.^[31] This normal subgroup lattice is always modular.^[31] Modularity means that for any normal subgroups L \subseteq K and H, the identity L \vee (H \wedge K) = (L \vee H) \wedge K holds, where \vee denotes join and \wedge denotes meet.^[31] This property inherits from the broader subgroup lattice but applies specifically to normals due to their closure under conjugation.^[32] In the special case of abelian groups, where all subgroups are normal, the lattice is distributive; for example, in the infinite cyclic group \mathbb{Z}, the normal subgroups are precisely the subgroups n\mathbb{Z} for n \geq 0, forming a chain under inclusion: \mathbb{Z} \supseteq 2\mathbb{Z} \supseteq 4\mathbb{Z} \supseteq \dots, which is a distributive lattice.^[33] By the correspondence theorem, the normal subgroups of G containing a fixed normal subgroup N are in bijective correspondence with the normal subgroups of the quotient group G/N, preserving the lattice structure under inclusion.^[34] Concrete examples illustrate the structure. In the symmetric group S_3, the normal subgroups are the trivial subgroup \{e\}, the alternating subgroup A_3 of index 2, and S_3 itself, forming a chain lattice of length 2.^[35] A subnormal series of G is a chain of subgroups where each is normal in the previous one, providing a path in the normal subgroup lattice from G to the trivial subgroup; such series connect to more advanced concepts like composition series.^[36]

Quotients and Homomorphisms

Quotient Groups

If N is a normal subgroup of a group G, the quotient group G/N is defined as the set of all left cosets of N in G, equipped with the binary operation (gN)(hN) = ghN for g, h \in G.^[37]^[38] This operation is well-defined, meaning it does not depend on the choice of representatives g and h from their respective cosets, precisely because N is normal. To see this, suppose g' = gn and h' = hm for some n, m \in N; then (g'N)(h'N) = (gn)(hm)N = gnhmN = g(h(h^{-1}nh)m)N = gh((h^{-1}nh)m)N. Since N is normal, h^{-1}nh \in N, and (h^{-1}nh)m \in N as N is a subgroup, so this equals ghN.^[37]^[5] The set G/N forms a group under this operation. Associativity follows from that of G: ((gN)(hN))(kN) = (ghN)(kN) = ghkN = gN(hN(kN)). The identity element is the coset N, since gN \cdot N = gN = N \cdot gN. Inverses exist as (gN)^{-1} = g^{-1}N, because gN \cdot g^{-1}N = gg^{-1}N = N and similarly for the other side.^[37]^[38] The order of the quotient group satisfies |G/N| = [G : N] = |G|/|N|, the index of N in G. For example, taking N = n\mathbb{Z} in the additive group \mathbb{Z} yields the cyclic group \mathbb{Z}/n\mathbb{Z} of order n. Another instance is the quotient S_3 / A_3 \cong \mathbb{Z}/2\mathbb{Z}, where S_3 is the symmetric group on three letters and A_3 its alternating subgroup of order 3, so the quotient has order 2.^[37]^[38] If N \leq H \trianglelefteq G with N normal in G, then H/N is a normal subgroup of G/N, and the quotient (G/N)/(H/N) is isomorphic to G/H. This is a preview of the third isomorphism theorem. The first isomorphism theorem states that for a homomorphism \phi: G \to K, the quotient G / \ker \phi is isomorphic to the image \operatorname{im} \phi, connecting quotients directly to homomorphisms (with kernel details addressed separately).^[37]^[38] Subgroups that are not normal fail to produce quotient groups because the coset multiplication is not well-defined. For instance, in S_3, the subgroup H = \langle (1\ 2) \rangle is not normal; consider the product of cosets H \cdot (1\ 3)H: using representatives e and (1\ 3) gives (1\ 3)H, but using (1\ 2) from H and (1\ 3) gives (1\ 2)(1\ 3)H = (1\ 3\ 2)H, and (1\ 3\ 2)H = \{(1\ 3\ 2), (1\ 3)\} \neq \{(1\ 3), (1\ 2\ 3)\} = (1\ 3)H.^[37]^[5]

Kernels of Homomorphisms

In group theory, the kernel of a group homomorphism plays a central role in connecting homomorphisms to normal subgroups. Given a homomorphism \phi: G \to H between groups G and H, the kernel \ker \phi is defined as the set \{g \in G \mid \phi(g) = e_H\}, where e_H is the identity element in H. This set forms a subgroup of G, and moreover, it is always normal in G. To see this, note that for any g \in G and n \in \ker \phi, the conjugate g n g^{-1} satisfies \phi(g n g^{-1}) = \phi(g) \phi(n) \phi(g)^{-1} = \phi(g) e_H \phi(g)^{-1} = e_H, so g n g^{-1} \in \ker \phi, confirming normality.^[39] The image of \phi, denoted \operatorname{im} \phi = \{\phi(g) \mid g \in G\}, is a subgroup of H. However, \operatorname{im} \phi is not necessarily normal in H. The cokernel of \phi is defined as the quotient H / \operatorname{im} \phi when \operatorname{im} \phi is normal in H; in general, for non-abelian groups, the cokernel may not exist unless this normality condition holds.^[40] A key result linking kernels, quotients, and homomorphisms is the third isomorphism theorem. Suppose N \trianglelefteq G is a normal subgroup of G, and consider a homomorphism \phi: G/N \to K. The kernel \ker \phi then corresponds to a subgroup M/N where N \leq M \trianglelefteq G, yielding an isomorphism (G/N) / (\ker \phi) \cong \operatorname{im} \phi. More precisely, if N \leq M \trianglelefteq G, then (G/N) / (M/N) \cong G/M. This theorem follows from the first isomorphism theorem applied to the composition of the natural projection G \to G/N with \phi.^[41] The quotient G / \ker \phi satisfies a universal property with respect to homomorphisms from G. Specifically, for any homomorphism \psi: G \to K such that \ker \phi \subseteq \ker \psi, there exists a unique homomorphism \overline{\psi}: G / \ker \phi \to K such that \psi = \overline{\psi} \circ \pi, where \pi: G \to G / \ker \phi is the natural projection. This property characterizes the quotient as the "universal" way to factor out the kernel.^[42] Illustrative examples highlight these concepts. The sign homomorphism \operatorname{sgn}: S_n \to \{ \pm 1 \}, which maps a permutation to the sign of its corresponding permutation matrix (or equivalently, +1 for even permutations and -1 for odd), has kernel exactly the alternating group A_n, which is thus normal in S_n. Another example is the projection homomorphism \pi: \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}, sending an integer k to its residue class modulo n; here, \ker \pi = n\mathbb{Z}, the multiples of n, which is normal in \mathbb{Z}.^[43]^[44]

Advanced Structures

Normal Series and Composition Series

A subnormal series of a group G is a finite chain of subgroups G = N_0 \trianglerighteq N_1 \trianglerighteq \cdots \trianglerighteq N_k = \{e\}, where each N_{i+1} is a normal subgroup of N_i.^[45] The successive quotients N_i / N_{i+1} are called the factors of the series.^[45] A normal series is a subnormal series in which every N_i is normal in the whole group G. Any two subnormal series of a finite group admit common refinements whose factor groups are pairwise isomorphic up to permutation. This refinement property is known as the Schreier refinement theorem, which provides the foundation for the Jordan–Hölder theorem applied to composition series.^[45] A composition series is a subnormal series that cannot be refined further, meaning each factor N_i / N_{i+1} is a simple group. The Jordan–Hölder theorem states that any two composition series of a finite group have the same length and the same simple factors up to isomorphism and permutation. A solvable series is a subnormal series whose factors N_i / N_{i+1} are all abelian groups.^[46] For example, the symmetric group S_3 admits the solvable series S_3 \trianglerighteq A_3 \trianglerighteq \{e\}, where A_3 is the alternating subgroup of order 3, yielding factors S_3 / A_3 \cong \mathbb{Z}/2\mathbb{Z} and A_3 / \{e\} \cong \mathbb{Z}/3\mathbb{Z}, both abelian.^[46] A chief series is a maximal normal series (with all subgroups normal in G), meaning there are no further G-normal subgroups between consecutive terms, and its factors (chief factors) are minimal normal subgroups of the corresponding quotients, which for finite groups are characteristically simple (direct products of isomorphic simple groups, often elementary abelian p-groups).^[47] In nilpotent groups, the lower central series provides a canonical example of a normal series. Defined recursively by \gamma_0(G) = G and \gamma_{i+1}(G) = [G, \gamma_i(G)], where [G, H] is the commutator subgroup generated by elements ghg^{-1}h^{-1} for g \in G and h \in H, this series terminates at the trivial subgroup after finitely many steps.^[15] Each \gamma_i(G) is normal in G, and the factors \gamma_i(G) / \gamma_{i+1}(G) are abelian (actually nilpotent of class at most 1 less).^[15]

Simple Groups and Solvability

A simple group is a nontrivial group whose only normal subgroups are the trivial subgroup and the group itself.^[48] This property implies that simple groups have no proper nontrivial normal subgroups, making them the "atoms" or building blocks in the structure of finite groups via composition series. Representative examples include cyclic groups of prime order, which are abelian simple groups, the alternating group A_5 of order 60, which is the smallest non-abelian simple group, and the projective special linear group \mathrm{PSL}(2,7) of order 168.^[49] The classification of finite simple groups, completed in 2004, states that every finite simple group is isomorphic to one of 26 sporadic groups, an alternating group A_n for n \geq 5, a group of Lie type (such as \mathrm{[PSL](/page/PSL)}(2,q) for certain q), or a cyclic group of prime order.^[50] This monumental result, spanning thousands of pages across multiple volumes, provides a complete list and underscores the rarity and structured nature of simple groups.^[51] A group G is solvable if it possesses a subnormal series \{H_i\} with H_0 = G \triangleright H_1 \triangleright \cdots \triangleright H_k = \{e\} such that each factor group H_i / H_{i+1} is abelian.^[52] In the context of Galois theory, a polynomial is solvable by radicals if and only if the Galois group of its splitting field over the rationals is solvable, linking group-theoretic solvability directly to the constructibility of roots via field extensions.^[53] For instance, the alternating group A_5 is non-solvable because it is a non-abelian simple group, so its derived series G^{(0)} = A_5, G^{(1)} = [A_5, A_5] = A_5, and G^{(k)} = A_5 for all k \geq 1 stabilizes at the nontrivial group itself rather than the trivial subgroup.^[54] A perfect group G satisfies G = G', where G' is the derived (commutator) subgroup, meaning G has no nontrivial abelian quotients.^[55] Non-abelian simple groups are perfect because their derived subgroup G' is a nontrivial proper normal subgroup (since G is non-abelian), but the only such subgroups are \{e\} and G, forcing G' = G. Burnside's theorem from 1904 advanced the detection of normal subgroups in p-groups by providing conditions under which groups of order p^a q^b (for distinct primes p, q) possess nontrivial normal Sylow subgroups, thereby ruling out non-abelian simple groups of such orders and facilitating solvability proofs.^[56]

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[PDF] Modern Algebra
Let H and K be finite subgroups of a group G. Then. |HK| = |H||K|/|H ∩ K|. Proof. Let C = H ∩ K. Then C is a finite subgroup of G by Corollary. I.2.6. C is ...
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Normal subgroups
A normal subgroup is the union of conjugacy classes. A subgroup H⊆G H ⊆ G is normal if and only if it is the union of conjugacy classes. 🔗. Proof. This is ...
[29]
[PDF] quotient groups - keith conrad
In every group G both G and its trivial subgroup {e} are normal subgroups of G. Example 2.5. If G is an abelian group then every subgroup of G is a normal ...
[30]
[PDF] On Covering Groups With Proper Subgroups - BearWorks
For this problem, we first need what is called the normal core of a subgroup. Let G be a group with subgroup A. Let G act on the set of left cosets of A, de ...
[31]
[PDF] Using the Maple Computer Algebra System as a Tool for Studying ...
The normal closure of a subgroup H of a group G is the smallest normal subgroup of G that contains H. By proposition 3.3, a subgroup H of a group G is ...
[32]
[PDF] 16. Characteristic subgroups and Products - UCSD Math
It turns out that most of the general normal subgroups that we have defined so far are all in fact characteristic subgroups.
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normal subgroup lattice is modular - PlanetMath.org
Mar 22, 2013 · The fact that the normal subgroups of a group G G form a lattice (call it N(G) N ⁢ ( G ) ) is proved here ...
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normal subgroup in nLab
### Summary of Lattice of Normal Subgroups from https://ncatlab.org/nlab/show/normal+subgroup
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https://mathweb.ucsd.edu/~atparris/small_groups.html
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Groups of small order. - UCSD Math
The first three proper subgroups have order two, while <s> has order three and is the only normal one. The center of S_3 is trivial.
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subnormal series - PlanetMath
Mar 22, 2013 · A subnormal series in which each Gi is a maximal normal subgroup of Gi−1 G i - 1 is called a composition series .
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https://users.math.msu.edu/users/meierfra/classnotes/m411s06/m411s06notes.pdf
[39]
[PDF] MTH 411 Lecture Notes Based on Hungerford, Abstract Algebra
Aug 28, 2014 · So N is normal subgroup if and only if (i)-(iv) hold. The phrase ... G/N is called the quotient group of G with respect to N. Theorem ...
[40]
[PDF] 18.703 Modern Algebra, Homomorphisms and kernels
Let φ: G -→ H be a homomorphism. Then the kernel of φ is a normal subgroup of G. Proof. We have already seen that the kernel is a subgroup. Suppose.
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[PDF] homomorphisms.pdf - Keith Conrad
Let f : G → H be a homomorphism of groups. (1) The image of a subgroup of G is a subgroup of H. In particular, f(G) is a subgroup of H. (2) The inverse ...
[42]
[PDF] 18.703 Modern Algebra, The Isomorphism Theorems
It is easy to prove the Third isomorphism Theorem from the First. Theorem 10.4 (Third Isomorphism Theorem). Let K ⊂ H be two normal subgroups of a group G.
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[PDF] The Universal Property of the Quotient
Mar 19, 2018 · Therefore, ˜φ is a homomorphism. The universal property of the quotient is an important tool in constructing group maps: To define a map out of ...
[44]
[PDF] 21 Conjugacy Classes for Symmetric and Alternating Groups
Today, we will be looking at the conjugacy classes for Sn and An, the symmetric group and the alternating ... normal subgroup. In particular, a normal ...
[45]
[PDF] MATH 415 Modern Algebra I Lecture 10: Homomorphisms of groups.
Examples of homomorphisms: • Residue modulo n of an integer. For any k ∈ Z let f (k) be the remainder of k after division by n.
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[PDF] SUBGROUP SERIES I 1. Introduction If N is a nontrivial proper ...
A normal series for a group G is a series of subgroups where each subgroup is normal in the next larger subgroup, such as {e} = G0 < G1 < G2 < ... < Gr = G.
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[PDF] Definition 1.10. If S is a subset of a group G then the subgroup ...
Here A3 = {e,(123),(132)} is the alternating group. This is a cyclic group and thus abelian and S3/A3 ∼= Z/2 is also abelian. So, S3 is solvable of degree 2. ( ...
[48]
[PDF] Chief factors - University of Kentucky
Mar 5, 2008 · Chief series. A chief series of a finite group G is a chain of G-normal subgroups 1 = H0 < H1 <...< Hn = G such that there are no. G-normal ...
[49]
[PDF] Chapter 25 Finite Simple Groups
Definition A group is simple if it has no nontrivial proper normal subgroup. The definition was proposed by Galois; he showed that An is simple for.
[50]
[PDF] Lecture 5.7: Finite simple groups
Every finite simple group is isomorphic to one of the following groups: A cyclic group Zp, with p prime;. An alternating group An, with n ≥ 5;. A Lie-type ...
[51]
Gorenstein, Lyons, and Solomon: The Classification of the Finite ...
Daniel Gorenstein, Richard Lyons, and Ronald Solomon, Mathematical Surveys and Monographs, vol. 40, The Classification of the Finite Simple Groups, Number 1.
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Simple groups classification - MacTutor History of Mathematics
When the Classification of Finite Simple Groups was completed by Michael Aschbacher and Stephen Smith in 2004, Aschbacher wrote the article 'The Status of ...
[53]
AATA Solvable Groups - Abstract Algebra: Theory and Applications
Proof. ... A group G is solvable if it has a subnormal series { H i } such that all of the factor groups H i + 1 / H i are abelian. Solvable groups will play a ...
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[PDF] Galois Theory and Solvability - Whitman People
Feb 2, 2023 · ... Galois group is solvable. In particular, if the Galois group is Sn, n ≥ 5, then the roots are individually not expressible by radicals.
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(PDF) A New Proof for Unsolvability of A5 - ResearchGate
May 9, 2020 · PDF | On May 9, 2020, Duy Nguyen published A New Proof for Unsolvability of A5 | Find, read and cite all the research you need on ...
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Perfect Group -- from Wolfram MathWorld
A group that coincides with its commutator subgroup. If G is a non-Abelian group, its commutator subgroup is a normal subgroup other than the trivial group.
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ON BURNSIDE'S OTHER paqh THEOREM - Project Euclid
In a less famous theorem, Burnside gave sufficient conditions for G to have a nontrivial normal p-subgroup for a particular prime p.