# Identification of groups given by a description

In a classification of groups of order $$p^3q$$, where $$p, q$$ are distinct primes, when considering non-nilpotent groups with normal Sylow $$p$$-subgroups they mention that these groups have the form $$N \rtimes U$$, with $$|N|=p^3$$ and $$U=C_q$$. Then there is a case where,

1) $$N$$ is extraspecial of exponent $$p$$: Then $$Aut(N) \rightarrow Aut(N/ \phi(N)) \cong GL(2,p)$$ is surjective. Thus the conjugacy classes of subgroups of order $$q$$ in $$Aut(N)$$ correspond to the conjugacy classes of subgroups of order $$q$$ in $$GL(2,p)$$.

and,

2) $$N$$ is extraspecial of exponent $$p^2$$: Then $$Aut(N)$$ is solvable and has a normal Sylow $$p$$-subgroup with $$p$$-complement of the form $$C_{p-1}$$. Thus there is atmost one subgroup of order $$q$$ in $$Aut(N)$$ and this exists iff $$q|(p-1)$$.

Can someone please help me to identify how to write the above two groups in notation form. i.e. Can the groups be written as follows using notations:

1) $$N \cong C_{p^2} \times C_p$$

2) $$N \cong C_{p^3}$$

• You'd need to understand first extraspecial groups though. No? The examples you give for the iso. type of $N$ are abelian groups. – the_fox Jan 31 at 4:17
• Yes, thanks @the_fox. But is it possible to mention the groups for the moment. I started to read about extraspecial groups. But please can you mention the groups? – Buddhini Angelika Jan 31 at 4:42
• The exponent-$p$ type has shape $(C_p \times C_p) \rtimes C_p$ and the exponent-$p^2$ type has shape $C_{p^2} \rtimes C_p$. I guess this is what you are asking. You should learn about these groups though, as they are quite important! – the_fox Jan 31 at 5:12
• Yes, thank you very much @the_fox. – Buddhini Angelika Jan 31 at 5:15
• Sorry. the answer was not loaded in my browser earlier and I have read it bit differently. But now its properly loaded and is clear. Thank you very much. :) – Buddhini Angelika Jan 31 at 5:17

An extraspecial $$p$$-group is a nonabelian group $$N$$ such that the center $$Z(N)$$ is cyclic of order $$p$$ and $$N/Z(N)$$ is an elementary abelian $$p$$-group, i.e. it is isomorphic to $$C_p^n$$ for some $$n$$. So, an extraspecial group of exponent $$p$$ is an extraspecial $$p$$-group whose exponent is $$p$$, and an extraspecial group of exponent $$p^2$$ is an extraspecial $$p$$-group whose exponent is $$p^2$$. (These are the only possible exponents for an extraspecial $$p$$-group, since $$Z(N)$$ and $$N/Z(N)$$ both have exponent $$p$$ and so $$x^{p^2}=1$$ for all $$x\in N$$.)
The extraspecial groups can be completely classified; you can find more information at the link above. In particular, for $$p$$ odd, an extraspecial group of order $$p^3$$ and exponent $$p$$ is isomorphic to the group of matrices of the form $$\begin{pmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1\end{pmatrix}$$ over $$\mathbb{F}_p$$, and an extraspecial group of order $$p^3$$ and exponent $$p^2$$ is a semidirect product $$C_{p^2}\rtimes C_p$$ where the action of $$C_p$$ on $$C_{p^2}$$ is nontrivial.