On p-deficiency in groups

PRO-p GROUPS OF POSITIVE DEFICIENCY

Let Γ be a finitely presentable pro-p group such that def(Γ) > 0. If Γ has a nontrivial finitely generated closed normal subgroup N of infinite index then def(Γ) = 1, N is a free pro-p group and Γ/N is virtually free. If π is a finitely presentable group with β (2) 1 (π) = 0 then def(π) ≤ 1, with equality only if c.d.π = 2 or π ∼= Z. (See Theorem 2.5 of [1].) The L-Betti number condition holds ...

On the Structure of Some Deficiency Zero Groups

ON COMMUTATORS IN p-GROUPS

For a given prime p, what is the smallest integer n such that there exists a group of order p in which the set of commutators does not form a subgroup? In this paper we show that n = 6 for any odd prime and n = 7 for p = 2.

On Sushchansky p-groups

We study Sushchansky p-groups introduced in [Sus79]. We recall the original definition and translate it into the language of automata groups. The original actions of Sushchansky groups on p-ary tree are not level-transitive and we describe their orbit trees. This allows us to simplify the definition and prove that these groups admit faithful level-transitive actions on the same tree. Certain br...

On strongly flat p-groups

In this paper, we provide a survey (both theoretical and computational) of the conditions for determining which finite p -groups are strongly flat. Let m(G) denote the maximum size of an irredundant generating set of a group G . Then, a group G is strongly flat if m(G) is strictly greater than m(H) for all proper subgroups H of G . For example, J. Whiston has shown in his dissertation that all ...