# 26.1 More about Special Ag Groups

Now the properties of a special ag system are described. First of all the Leedham-Green series will be introduced.

Let G = G_1 > G_2 > ldots > G_m > G_{m+1} = { 1 } be the lower nilpotent series of G, i.e., G_i is the smallest normal subgroup of G_{i-1} such that G_{i-1} / G_i is nilpotent.

To refine this series the lower elementary abelian series of a nilpotent group N will be constructed. Let N = P_1 cdot ldots cdot P_l be the direct product of its Sylow-subgroups such that P_h is a p_h-group and p_1 < p_2 < ldots < p_l holds. Let lambda_{j}(P_h) be the j-th term of the p_h-central series of P_h and let k_h be the length of this series (see PCentralSeries). Define N_{j, p_h} as the subgroup of N with N_j, p_h = lambda_j+1(P_1) cdots lambda_j+1(P_h-1) cdot lambda_j(P_h) cdots lambda_j(P_l). With k = max{k_1, ldots, k_l} the series N = N_1, p_1 geq N_1, p_2 geq ldots geq N_1,p_l geq N_2, p_1 geq ldots geq N_k, p_l = { 1 } is obtained. Since the p-central series may have different lengths for different primes, some subgroups might be equal. The lower elementary abelian series is obtained, if for all pairs of equal subgroups the one with the lexicographically greater index is removed. This series is a characteristic central series with maximal elementary abelian factors.

To get the Leedham-Green series of G, each factor of the lower nilpotent series of G is refined by its lower elementary abelian series. The subgroups of the Leedham-Green series are denoted by G_{i, j, p_{i, h}} such that G_{i, j, p_{i, h}} / G_{i+1} = (G_i / G_{i+1})_{j, p_{i,h}} for each prime p_{i,h} dividing the order of G_i / G_{i+1}. The Leedham-Green series is a characteristic series with elementary abelian factors.

A PAG system corresponds naturally to a composition series of its group. The first additional property of a special ag system is that the corresponding composition series refines the Leedham-Green series.

Secondly, all the elements of a special ag system are of prime-power order, and furthermore, if a set of primes pi = {q_1, ldots, q_r} is given, all elements of a special ag system which are of q_h-power order for some q_h in pi generate a Hall-pi-subgroup of G. In fact they form a canonical generating sequence of the Hall-pi-subgroup. These Hall subgroups are called public subgroups, since a subset of the PAG system is an induced generating set for the subgroup. Note that the set of all public Sylow subgroups forms a Sylow system of G.

The last property of the special ag systems is the existence of public local head complements. For a nilpotent group N, the group lambda_2(N) = lambda_2(P_1) cdots lambda_2(P_l) is the Frattini subgroup of N. The local heads of the group G are the factors (G_i / G_i+1) / lambda_2(G_i / G_i+1) = G_i / G_i, 2,p_i,1 for each i. A local head complement is a subgroup K_i of G such that K_i / G_{i,2,p_{i,1}} is a complement of G_i / G_{i, 2, p_{i 1}}. Now a special ag system has a public local head complement for each local head. This complement is generated by the elements of the special ag system which do not lie in G_i \ G_{i,2,p_{i,1}}. Note that all complements of a local head are conjugate. The factors lambda_2(G_i / G_i+1) = G_i, 2,p_i,1 / G_i+1 are called the tails of the group G.

To handle the special ag system the weights are introduced. Let (g_1, ldots, g_n) be a special ag system. The triple (w_1, w_2, w_3) is called the weight of the generator g_i if g_i lies in G_{w_1, w_2, w_3} but not lower down in the Leedham-Green series. That means w_1 corresponds to the subgroup in the lower nilpotent series and w_2 to the subgroup in the elementary-abelian series of this factor, and w_3 is the prime dividing the order of g_i. Then weight(g_i) = (w_1, w_2, w_3) and weight_j(g_i) = w_j for j = 1,2,3 is set. With this definition {g_i | weight_3(g_i) in pi} is a Hall-pi-subgroup of G and {g_i | weight(g_i) neq (j, 1, p) mbox{ for some } p } is a local head complement.

Now some advantages of a special ag system are summarized.

item[1.] You have a characteristic series with elementary abelian factors of G explicitly given in the ag system. This series is refined by the composition series corresponding to the ag system.

item[2.] You can see whether G is nilpotent or even a p-group, and if it is, you have a central series explicitly given by the Leedham-Green series. Analogously you can see whether the group is even elementary abelian.

item[3.] You can easily calculate Hall-pi-subgroups of G. Furthermore the set of public Sylow subgroups forms a Sylow system.

item[4.] You get a smaller generating set of the group by taking only the elements which correspond to local heads of the group.

item[5.] The collection with a special ag system may be faster than the collection with an arbitrary ag system, since in the calculation of the public subgroups of G the commutators of the ag generators are shortened.

item[6.] Many algorithms are faster for special ag groups than for arbitrary ag groups.

GAP 3.4.4
April 1997