Extra special $p$-groups

Question

I am looking for a reference for the following theorem:

Let $G$ and $H$ be two extra special $p$-groups with the same order and the same exponent. Then $G\cong H$.

Thank you in advance

Answer 1

A reference is Theorem 5.2 of the book Finite Groups by Daniel Gorenstein (1980, Chelsea Publishing Company).

The theorem is the full description of extra special $p$-groups. For $m$ odd there are exactly two extra special $p$-groups of order $p^m$, one has exponent $p$ and the other has exponent $p^2$. There aren't any extra special $p$-groups of order $p^m$ when $m$ is even. So we conclude that if two extra special $p$-groups have the same order and exponent then they are isomorphic.

Here are some lecture notes that cover it in Chapter 3.

http://web.mat.bham.ac.uk/D.A.Craven/docs/lectures/pgroups.pdf