4
$\begingroup$

I have recently found, as part of a problem in finite groups, the following abstract groups (where $p$ is an odd prime):

$G_1 = \langle \tau, x, y : \tau^2 = x^p = y^p = 1, xy = yx, \tau x \tau^{-1} = x, \tau y \tau^{-1} = y^{-1} \rangle$ $G_2 = \langle \tau, x, y : \tau^2 = x^p = y^p = 1, xy = yx, \tau x \tau^{-1} = x^{-1}, \tau y \tau^{-1} = y^{-1} \rangle$

I am wondering if there is a common name for groups with either of these structures, so that I can look up existing results about them. I have never seen them in my undergraduate studies so far,

  • 0
    I don't know about any names, but both can be written as semi-direct products of cyclic groups. – 2017-01-01
  • 1
    I'm not completely clear on semi-direct products, how would that work? I would guess that $G_2 = C_2 \rtimes (C_p \times C_p)$? – 2017-01-01
  • 3
    You have the semidirect product symbol the wrong way round. It is the $C_p \times C_p$ that is normal, so it should be $C_2 \ltimes (C_p \times C_p)$ or $(C_p \times C_p) \rtimes C_2$. And both $G_1$ and $G_2$ have that structure. – 2017-01-01

2 Answers 2

4

There are two ways of writing these groups, although both of them will be ambiguous (since without any further indication it owuld be unclear which one is referring to).

The first one, as suggested by Arnaud D., is writing it as a semidirect product of cyclic groups. Recall that given groups $G$ and $H$ and a group homomorphism $\varphi:H\rightarrow G$, the semidirect product of $G$ and $H$ with respect to $\varphi$ is the group $G\rtimes_{\varphi}H$ whose underlying set is $G\times H$ and its product is given by $$(g_0,h_0)(g_1,h_1)=(g_0\varphi(h_0)(g_1),h_0h_1)\text{.}$$ Note that this generalizes the direct product in the sense that those are just the semidirect product where $\varphi$ is the trivial map (which sends every element to the identity). In this way, in sloppy notation, both of your groups would be $(C_p \times C_p) \rtimes C_2$. Note that $\rtimes$ is $\times$ together with $\triangleleft$, used to say that something is a normal subgroup, so that's why the order is reversed.

The second one I have seen is that of (sub)holomorph groups. Given a group $G$ and a subgroup $N$ of its automorphism group $\mathrm{Aut}(G)$, the $N$-holomorph group of $G$ is the group $$\mathrm{Hol}(G,N):=G\rtimes_{\iota} N$$ where $\iota$ is the inclusion map. This is less general and less common, as the holomorph group of $G$ refers usually to $\mathrm{Hol}(G):=\mathrm{Hol}(G,\mathrm{Aut}\,G)$. Using sloppy notation, your groups would be written as $\mathrm{Hol}(C_p\times C_p,C_2)$.

  • 0
    https://www.researchgate.net/project/Character-degrees-of-Finite-Groups – 2017-01-01
3

$G_1$ is the direct product of a cyclic group and a dihedral group, $C_p\times D_p$ (or $C_p\times D_{2p}$ depending on your notation).

$G_2$ is the generalised dihedral group on the abelian group $C_p^2$.

(See https://en.wikipedia.org/wiki/Generalized_dihedral_group or https://groupprops.subwiki.org/wiki/Generalized_dihedral_group)