Prime-Power Subgroups, Composition Series and Solvability

Let p be a prime. A maximal p-subgroups of a finite group G is a maximal p-subgroup P. Maximality means that if Q is a p-subgroup of G and P ≤Q, then P = Q. It follows from subgroup-order theorem that if pe is the largest power of p dividing |G|, then a subgroup of order pe, should it exist, is a maximal p-subgroup of G. One virtue of the present definition is that maximal p-subgroups always exist: indeed, we now show that if S is any p-subgroup of G (perhaps S = {1}), then there exists a maximal p-subgroups P containing S. If there is no p-subgroup strictly containing S, then S itself is a maximal p-subgroups. Otherwise, there is a p-subgroup P1 with S < P1. If P1 is maximal, it is prime-power subgroup, and we are done. Otherwise, there is some p-subgroup P2 with P1 < P2. This procedure of producing larger and larger p-subgroups Pi must end after a finite number of steps, for |Pi| ≤|G| for all i; the largest Pi must, therefore, be a maximal p-subgroups. Recall that a conjugate of a subgroup H ≤G is a subgroup of G of the form aHa−1 = {aha−1 : h ∈H}, where a ∈G. The normalizer of H in G is the subgroup NG(H) = {a ∈G : aHa−1 = H}, and Proposition 2.101 states that if H is a subgroup of a finite group G, then the number of conjugates of H in G is [G : NG(H)]. It is obvious that H ✁NG(H), and so the quotient group NG(H)/H is defined.

A unitriangular matrix over a field k is an upper triangular matrix each of whose diagonal terms is 1. Define UT(n, k) to be the set of all n ×n unitriangular matrices over k. Remark. We can generalize this definition by allowing k to be any commutative ring. For example, the group UT(n, Z) is an interesting group. ◀

A composition series is a normal series all of whose nontrivial factor groups are simple. The nontrivial factor groups of a composition series are called composition factors of G. A group need not have a composition series; for example, the abelian group Z has no composition series. However, every finite group does have a composition series.

Two normal series of a group G are equivalent if there is a bijection between the sets of nontrivial factor groups of each so that corresponding factor groups are isomorphic. The composition-factor theorem says that any two composition series of a group are equivalent.

If G is a group and K ✁G, then G is called an extension of K by G/K. Holder proved that the factor groups themselves, to isomorphism, do not depend upon the composition series. The study of extensions involves the inverse question: How much of G can be recovered from a normal subgroup K and the quotient Q = G/K? For example, we do know that if K and Q are finite, then |G| = |K||Q|.

If G is a group and x, y ∈G, then their commutator [x, y] is the element [x, y] = xyx−1y−1. If X and Y are subgroups of a group G, then [X, Y] is defined by [X, Y] = [x, y] : x ∈X and y ∈Y . In particular, the commutator subgroup G′ of a group G is G′ = [G, G], the subgroup generated by all the commutators.5 It is clear that two elements x and y in a group G commute if and only if their commutator [x, y] is 1. The next proposition generalizes this observation. 5The subset consisting of all the commutators need not be closed under products, and so the set of all commutators may not be a subgroup. The smallest group in which a product of two commutators is not a commutator has order 96. The composition-factor Theorem

The derived series of G is G = G(0) ≥G(1) ≥G(2) ≥· · · ≥G(i) ≥G(i+1) ≥· · · , where G(0) = G, G(1) = G′, and, more generally, G(i+1) = (G(i))′ = [G(i), G(i)] for all i ≥0. It is easy to prove by induction on i ≥0, that G(i) is fully invariant, which implies that G(i) ✁G; it follows that G(i+1) ✁G(i), and so the derived series is a normal series. The derived series can be used to give a characterization of solvability: G is solvable if and only if the derived series reaches {1}.

The descending central series of a group G is G = γ1(G) ≥γ2(G) ≥· · · , where γi+1(G) = [γi(G), G]. A group G is called nilpotent if the lower central series reaches {1}; that is, if γn(G) = {1} for some n. Note that γ2(G) = G′, but the derived series and the lower central series may differ afterward; for example, γ3(G) = [G′, G] ≥G(2), with strict inequality possible. Finite nilpotent groups can be characterized by Proposition 5.39: they are the groups that are direct products of their maximal prime-power subgroups, and so one regards finite nilpotent groups as generalized p-groups.

Let P be a maximal p-subgroups of a finite group G. (i) Every conjugate of P is also a maximal p-subgroups of G. (ii) |NG(P)/P| is prime to p. (iii) If a ∈G has order some power of p and if aPa−1 = P, then a ∈P.

Let G be a finite group of order pe1 1 · · · pet t , and let P be a maximal p-subgroups of G for some prime p = p j. (i) Every maximal p-subgroups is conjugate to P. The maximal p-subgroup theorems (ii) If there are r j p j-subgroups, then r j is a divisor of |G|/p e j j and r j ≡1 mod p j.

If G is a finite group of order pem, where p is a prime and p ∤m, then every maximal p-subgroups P of G has order pe.

If G is a finite group of order pem, where p is a prime and p ∤m, then G has a subgroup of order pe.

There is no nonabelian simple group G of order |G| = pem, where p is prime, p ∤m, and pe ∤(m −1)!.

says that G is not simple unless |G| = p. Suppose that such a simple group G exists. By maximal p-subgroup theorem, G contains a subgroup P of order pe, hence of index m. We may assume that m > 1, for nonabelian p-groups are never simple. By Theorem 2.88, there exists a homomorphism ϕ : G →Sm with ker ϕ ≤ P. Since G is simple, however, it has no proper normal subgroups; hence ker ϕ = {1} and ϕ is an injection; that is, G ∼= ϕ(G) ≤Sm. By subgroup-order theorem, pem | m!, and so pe | (m −1)!, contrary to the hypothesis. •

Let q = pe, where p is a prime. For each n ≥2, UT(n, Fq) is a p-group of order q(n 2) = qn(n−1)/2.

If p is an odd prime, then there exists a nonabelian group G of order p3 with x p = 1 for all x ∈G.

If p is a prime and q = pm, then the unitriangular group UT(n, Fq) is a maximal p-subgroups of GL(n, Fq).

If p is a prime and G is a finite p-group, then G is isomorphic to a subgroup of the unitriangular group UT(|G|, Fp).

Any two normal series G = G0 ≥G1 ≥· · · ≥Gn = {1} and G = N0 ≥N1 ≥· · · ≥Nk = {1} of a group G have equivalent refinements.

Any two composition series of a group G are equivalent. In particular, the length of a composition series, if one exists, is an invariant of G.

Let G be a group. (i) The commutator subgroup G′ is a normal subgroup of G, and G/G′ is abelian. (ii) If H ✁G and G/H is abelian, then G′ ≤H.

(i) A finite group G is solvable if and only if it has a normal series with abelian factor groups. (ii) A finite group G is solvable if and only if there is some n with G(n) = {1}.

(i) Let G = S4. Now |S4| = 24 = 233. Thus, a maximal 2-subgroup of S4 has order 8. The maximal p-subgroup theorems says that all subgroups of order 8 are conjugate, hence isomorphic, to D8. Moreover, the number r of prime-power subgroup 2subgroups is a divisor of 24 congruent to 1 mod 2; that is, r is an odd divisor of 24. (ii) If G is a finite abelian group, then a maximal p-subgroups is just its p-primary component (since G is abelian, every subgroup is normal, and so there is a unique maximal p-subgroups for every prime p).

(i) A group G is abelian if and only if G′ = {1}. (ii) If G is a simple group, then G′ = {1} or G′ = G, for G′ is a normal subgroup. The first case occurs when G has prime order; the second case occurs otherwise. In particular, (An)′ = An for all n ≥5. (iii) We show that (Sn)′ = An for all n ≥5. Since Sn/An ∼= I2 is abelian, Proposition 5.57 shows that (Sn)′ ≤An. For the reverse inclusion, note that (Sn)′ ∩An ✁An, so that the simplicity of An gives this intersection trivial or An. Clearly, (Sn)′ ∩An ̸= {(1)}, and so An ≤(Sn)′. ◀ Let us iterate the formation of the commutator subgroup.