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I am trying to prove that if $G$ is a $p$-group of maximal class and order $p^4$ ($p$ odd), then its unique two-step centraliser $G_1=C_G([G,G])$ is of the form $C_{p^2}\times C_p$. It is clear from B …

The definition 1.7 from Endliche Gruppen, B.Huppert, vol-I, Chap.II, Pg.148 is as follows: Die Permutationsgruppe $\mathfrak G$ auf der Ziffernmenge $\Omega$ heißt $k$-fach transitiv $(k \leq |\Omega …

I am trying to prove the following comment (Ref. https://link.springer.com/article/10.1007/s00605-016-0938-5 Page-684, Rmk3.2): Let $G$ be a finite $p$-group of co-class $3$, class $\geq 4$. Then $G$ …

The well known Ore's conjecture (now established) states that every element of a finite non-abelian simple group $G$ is a commutator of a pair of elements. Also we know that $G$ is $2$-generated. I a …

I am trying to understand the classical method of counting classes from Burnside's old book (Note E) (also clarified a bit by Vera-Lopez, Conjugacy classes in finite solvable groups, 1984) : $G$ is a …

I am trying to look for a finite $p$-group of maximal class of order at least $p^{2p+1}$ exponent at least $p^3$ which possibly has the following property : If s is an element in $G-G_1$ ($G_1$ is th …

I am trying to look for the $2$-generated groups of order $3^7$ and class $4$ all whose upper central series quotients are elementary abelian of order 9 except the center which has order $3$. A small …

This is a very general question: are there known examples of Ramanujan behaviour of Cayley graphs obtained from family of finite p-groups? ${\mathrm{\bf Adjacency~matrix:}}$ Given a graph ${\mathcal{G …