We give bounds for the order of the Schur multiplier of a group of maximal class and order p(n ) for all primes p > 2, and we give examples to show that our bounds are best possible. In particular, for every p > 2 and n >= 3, we construct groups where the orders of their Schur multipliers attain our bounds. We describe the Schur multipliers of the 3-groups of maximal class, and we compute the Schur multipliers of the groups of maximal class of order p( )(n)for all p > 2 and all n <= 7.
For a finite group G, the vertices of the prime graph Gamma(G) are the primes that divide vertical bar G vertical bar, and two vertices p and q are connected by an edge if there is an element of order pq in G. Prime graphs of solvable groups have been classified, and prime graphs of groups whose noncyclic composition factors are isomorphic to a single nonabelian simple group T have been classified in the case where T has order divisible by exactly three or four distinct primes, except for the cases T = Sz(8) T D Sz(32), and T D PSL(2, q), which in some sense are the hardest cases. In this paper, we complete the classification for T = Sz(32), T = Sz(8), and T = PSL(2, 2(5)), with the latter two being the first cases ever studied where vertical bar Out(T)vertical bar has prime factors which do not divide vertical bar T vertical bar. The groups studied in this paper are also the first ones requiring knowledge of their Brauer character tables to complete the classification task.
Abstract Let G 1 G_{1} , G 2 G_{2} , … be a sequence of almost simple groups and construct a sequence ( W i ) (W_{i}) of wreath products via W 1 = G 1 W_{1}=G_{1} and, for each i > 1 i>1 , W i + 1 = G i + 1 wr W i W_{i+1}=G_{i+1}\mathbin{\mathrm{wr}}W_{i} via the regular action of each G i G_{i} . We determine the minimum number d ( W i ) d(W_{i}) of generators required for each wreath product in this sequence.
Abstract The classification of 𝑝-groups of maximal class is a deep and long-standing problem in group theory. Recently, a new approach was introduced by Eick, Komma and Saha: this uses the Lazard correspondence to construct the so-called frame 𝑝-groups . The set of all frame 𝑝-groups forms a significant, but still controllable part of the set of all 𝑝-groups of maximal class and hence seems a good basis for a classification. This paper investigates the automorphism groups and the isomorphism problem for frame 𝑝-groups.
Abstract We show that every convex subgroup of a left-ordered group is concentrated. This resolves a question of Bowditch who asked if every maximal cyclic subgroup of a free group is concentrated within his study of diffuse groups. We also express concentrated as a form of convexity in the Chiswell–Promislow framework of locally invariant order which is equivalent to diffuse.
