ABSTRACT ALGEBRA ON LINE: Structure of Groups

7.1.2. Theorem. [Second Isomorphism Theorem] Let G be a group, let N be a normal subgroup of G, and let H be any subgroup of G. Then HN is a subgroup of G, H

N is a normal subgroup of H, and

(HN) / N H / (H N).

7.1.3. Theorem. Let G be a group with normal subgroups H, K such that HK=G and H

K={e}. Then

G H × K .

7.1.4. Proposition. Let G be a group and let a G. The function i_a : G -> G defined by i_a(x) = axa^-1 for all x G is an isomorphism.

7.1.5. Definition. Let G be a group. An isomorphism from G onto G is called an automorphism of G. An automorphism of G of the form i_a, for some a G, where i_a (x) = axa^-1 for all x G, is called an inner automorphism of G. The set of all automorphisms of G will be denoted by Aut(G) and the set of all inner automorphisms of G will be denoted by Inn(G).

7.1.6. Proposition. Let G be a group. Then Aut(G) is a group under composition of functions, and Inn(G) is a normal subgroup of Aut(G).

7.1.7. Definition. For any group G, the subset

Z(G) = { x G | xg = gx for all g G }

is called the center of G.

7.1.8. Proposition. For any group G, we have Inn(G) G/Z(G).

Example. 7.1.1. Aut(Z) Z₂ and Inn(Z) = {e}

Example. 7.1.2. Aut(Z_n) Z_n^×

Conjugacy

7.2.1. Definition. Let G be a group, and let x,y

G. The element y is said to be a conjugate of the element x if there exists an element a

G such that y = axa^-1.
If H and K are subgroups of G, then K is said to be a conjugate subgroup of H if there exists a

G such that K = aHa^-1.

7.2.2. Proposition.

(a) Conjugacy of elements defines an equivalence relation on any group G.

(b) Conjugacy of subgroups defines an equivalence relation on the set of all subgroups of G.

7.2.3. Definition. Let G be a group. For any element x G, the set

{ a G | axa^-1 = x }

is called the centralizer of x in G, denoted by C(x).
For any subgroup H of G, the set

{ a G | aHa^-1 = H }

is called the normalizer of H in G, denoted by N(H).

7.2.4. Proposition. Let G be a group and let x G. Then C(x) is a subgroup of G.

7.2.5. Proposition. Let x be an element of the group G. Then the elements of the conjugacy class of x are in one-to-one correspondence with the left cosets of the centralizer C(x) of x in G.

Example. 7.2.3. Two permutations are conjugate in S_n if and only if they have the same shape (i.e., the same number of disjoint cycles, of the same lengths). Thus, in particular, cycles of the same length are always conjugate.

7.2.6. Theorem. [Conjugacy class Equation] Let G be a finite group. Then

| G | = | Z(G) | + [ G : C(x) ]

where the sum ranges over one element x from each nontrivial conjugacy class.

7.2.7. Definition. A group of order pⁿ, with p a prime number and n 1, is called a p-group.

7.2.8. Theorem. [Burnside] Let p be a prime number. The center of any p-group is nontrivial.

7.2.9. Corollary. Any group of order p² (where p is prime) is abelian.

7.2.10. Theorem. [Cauchy] If G is a finite group and p is a prime divisor of the order of G, then G contains an element of order p.

Groups acting on sets

7.3.1. Definition. Let G be a group and let S be a set. A multiplication of elements of S by elements of G (defined by a function from G × S -> S) is called a group action of G on S provided for each x

(i) a(bx) = (ab)x for all a,b

G, and

(ii) ex = x for the identity element e of G.

7.3.2. Proposition. Let G be a group and let S be a set. Any group homomorphism from G into the group Sym(S) of all permutations of S defines an action of G on S. Conversely, every action of G on S arises in this way.

7.3.3. Definition. Let G be a group acting on the set S. For each element x S, the set

Gx = { s S | s=ax for some a G }

is called the orbit of x under G, and the set

G_x = { a G | ax = x }

is called the stabilizer of x in G. The set

S^G = { x S | ax = x for all a G }

is called the subset of S fixed by G.

7.3.4. Proposition. Let G be a group that acts on the set S, and let xS.

(a) The stabilizer G_x of x in G is a subgroup of G.

(b) There is a one-to-one correspondence between the elements of the orbit Gx of x under G and the left cosets of G_x in G.

7.3.5. Proposition. Let G be a finite group acting on the set S.

(a) The orbits of S (under the action of G) partition S.

(b) For any x S, | Gx | = [ G : G_x ].

7.3.6. Theorem. Let G be a finite group acting on the finite set S. Then

| S | = | S^G | + [ G : G_x ],

where

is a set of representatives of the orbits Gx for which | Gx | > 1.

7.3.7. Lemma. Let G be a finite p-group acting on the finite set S. Then

| S | | S^G | (mod p).

7.3.8. Theorem. [Cauchy] If G is a finite group and p is a prime divisor of |G|, then the number of solutions in G of the equation x^p = e is a multiple of p. In particular, G has an element of order p.

The Sylow theorems

7.4.1. Theorem. [First Sylow Theorem] Let G be a finite group. If p is a prime such that p^k is a divisor of |G| for some k

0, then G contains a subgroup of order p^k.

7.4.2. Definition. Let G be a finite group, and let p be a prime number. A subgroup P of G is called a Sylow p-subgroup of G if |P| = p^k for some integer k 1 such that p^k is a divisor of |G| but p^k+1 is not.

7.4.3. Lemma. Let G be a finite group with |G| = mp^k, where k 1 and m is not divisible by p. If P is a normal Sylow p-subgroup, then P contains every p-subgroup of G.

7.4.4. Theorem. [Second and Third Sylow Theorems] Let G be a finite group of order n, and let p be a prime number.

(a) All Sylow p-subgroups of G are conjugate, and any p-subgroup of G is contained in a Sylow p-subgroup.

(b) Let n = mp^k, with gcd(m,p)=1, and let s be the number of Sylow p-subgroups of G.
Then s | m and s 1 (mod p).

7.4.5. Proposition. Let p > 2 be a prime, and let G be a group of order 2p. Then G is either cyclic or isomorphic to the dihedral group D_p of order 2p.

7.4.6. Proposition. Let G be a group of order pq, where p > q are primes.

(a) If q is not a divisor of p-1, then G is cyclic.

(b) If q is a divisor of p-1, then either G is cyclic or else G is generated by two elements a and b satisfying the following equations:
a^p = e, b^q = e, ba = aⁿb
where n 1 (mod p) but n^q 1 (mod p).

Finite abelian groups

7.5.1. Theorem. A finite abelian group can be expressed as a direct product of its Sylow p-subgroups.

7.5.2. Lemma. Let G be a finite abelian p-group, let a G be an element of maximal order, and let b<a> be any coset of G/<a>. Then there exists d G such that d<a> = b<a> and <a> <d> = {e}.

7.5.3. Lemma. Let G be a finite abelian p-group. If <a> is a cyclic subgroup of G of maximal order, then there exists a subgroup H with G <a> × H.

7.5.4. Theorem. [Fundamental Theorem of Finite Abelian Groups] Any finite abelian group is isomorphic to a direct product of cyclic groups of prime power order. Any two such decompositions have the same number of factors of each order.

7.5.5. Proposition. Let G be a finite abelian group. Then G is isomorphic to a direct product of cyclic groups

Z_n₁ × Z_n₂ × · · · × Z_{n_k}

such that n_i | n_i-1 for i = 2,3, . . . ,k.

7.5.6. Corollary. Let G be a finite abelian group. If a G is an element of maximal order in G, then the order of every element of G is a divisor of the order of a.

7.5.8 Theorem. Let p be an odd prime, let k be a positive integer, and let n = p^k. Then Z_n^× is a cyclic group.

7.5.10. Theorem. If k 3, and n = 2^k, then Z_n^× is isomorphic to the direct product of a cyclic group of order 2 and a cyclic group of order 2^k-2.

7.5.11. Corollary. The group Z_n^× is cyclic if and only if n is of the form 2, 4, p^k, or 2p^k for an odd prime p.

In elementary number theory, an integer g is called a primitive root for the modulus n if Z_n^× is a cyclic group and [g]_n is a generator for Z_n^×. Corollary 7.5.11 determines which moduli n have primitive roots. The proof of Theorem 7.5.8 shows how to find a generator for Z_n^×, where n = p^k.

Solvable groups

7.6.1. Definition. The group G is said to be solvable if there exists a finite chain of subgroups
G = N₀

N₁

· · ·

N_n such that

(i) N_i is a normal subgroup in N_i-1 for i = 1,2, . . . ,n,

(ii) N_i-1 / N_i is abelian for i = 1,2, . . . ,n, and

(iii) N_n = {e}.

7.6.2. Proposition. A finite group G is solvable if and only if there exists a finite chain of subgroups
G = N₀ N₁ · · · N_n such that

(i) N_i is a normal subgroup in N_i-1 for i = 1,2, . . . ,n,

(ii) N_i-1 / N_i is cyclic of prime order for i = 1,2, . . . ,n, and

(iii) N_n = {e}.

7.6.3. Theorem. Let p be a prime number. Any finite p-group is solvable.

7.6.4. Definition. Let G be a group. An element g G is called a commutator if

g = aba^-1b^-1

for elements a,b G. The smallest subgroup that contains all commutators of G is called the commutator subgroup or derived subgroup of G, and is denoted by G'.

7.6.5. Proposition. Let G be a group with commutator subgroup G'.

(a) The subgroup G' is normal in G, and the factor group G/G' is abelian.

(b) If N is any normal subgroup of G, then the factor group G/N is abelian if and only if G' N.

7.6.6. Definition. Let G be a group. The subgroup (G' )' is called the second derived subgroup of G. We define G^(k) inductively as (G^(k-1))', and call it the k th derived subgroup

7.6.7. Theorem. A group G is solvable if and only if G⁽ⁿ⁾ = {e} for some positive integer n.

7.6.8. Corollary. Let G be a group.

(a) If G is solvable, then so is any subgroup or homomorphic image of G.

(b) If N is a normal subgroup of G such that both N and G/N are solvable, then G is solvable.

7.6.9. Definition. Let G be a group. A chain of subgroups
G = N₀ N₁ . . . N_n such that

(i) N_i is a normal subgroup in N_i-1 for i = 1,2, . . . ,n,

(ii) N_i-1 / N_i is simple for i = 1,2, . . . ,n, and

(iii) N_n = {e}

is called a composition series The factor groups N_i-1 / N_i are called the composition factors determined by the series.

7.6.10. Theorem. [Jordan-Holder] Any two composition series for a finite group have the same length. Furthermore, there exists a one-to-one correspondence between composition factors of the two composition series under which corresponding composition factors are isomorphic.

Simple groups

7.7.1. Lemma. If n

3, then every permutation in A_n can be expressed as a product of 3-cycles.

7.7.2. Theorem. The symmetric group S_n is not solvable for n 5.

7.7.3. Lemma. If n 4, then no proper normal subgroup of A_n contains a 3-cycle.

7.7.4. Theorem. The alternating group A_n is simple if n 5.

7.7.5. Definition. Let F be a field. The set of all n × n matrices with entries in F and determinant 1 is called the special linear group over F, and is denoted by SL_n(F).
The group SL_n(F) modulo its center is called the projective special linear group and is denoted by PSL_n(F).

7.7.6. Proposition. For any field F, the center of SL_n(F) is the set of nonzero scalar matrices with determinant 1.

Example. 7.7.1. PSL₂(F) S₃ if |F| = 2.

Example. 7.7.2. PSL₂(F) A₄ if |F| = 3.

7.7.7. Lemma. Let F be any field. Then SL₂(F) is generated by elements of the form
and .

7.7.8. Lemma. Let F be any finite field, and let N be a normal subgroup of SL₂(F). If N contains an element of the form with a 0, then N = SL₂(F).

7.7.9. Theorem. Let F be any finite field with |F| > 3. Then the projective special linear group PSL₂(F) is a simple group.

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