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Questions about the branch of algebra that deals with groups.

2 votes
0 answers
238 views

Questions about some finite p-groups of coclass 2

I have two questions about a specific type of finite $p$-groups that i've seen in an interesting work on automorphisms of finite p-groups. Let $G$ a finite $p$-group of order $p^{n}$ such that: - $ …
1 vote
1 answer
768 views

Groups in which lower central series and upper central series coincide

Let $G$ a finite two-generated $p$-group in which lower and upper central series coincide. Clearly we obtain that the upper central series become strongly central, we have also that at least half of …
0 votes
1 answer
500 views

Centralizer of derived subgroup

In all questions suppose $G$ metabelian p-group such that G is not regular ( so $cl(G) \geq p$ ), G is not a wreath product; $Z(G) \leq \phi(G)$. 1) Let $M$ normal abelian subgroup of $G$ such th …