What does $\frac12(D_{2p}\times D_{2p})$ mean in group theory?

Question

Reading a thesis, I have come across the (unexplained) notation $$\frac{1}{2}(D_{2p}\times D_{2p})\cong (p\times p):2,$$ where $D_{2p}$ is a dihedral group. What does this "$\frac12$" notation mean? For what groups $G$ does $\frac12 G$ make sense?

Answer 1

Fleshing out my comment above: the notation $\frac1f(G\times H)$ actually appears in the book On Quaternions and Octonions (chapter 4.3, "Naming the Groups"). The explanation there says:

$\frac1f[\![L\times R]\!]$ consists of the elements $[\![l,r]\!]$ for which $l\in 2L, r\in 2R$, and $l^\alpha=r^\beta$, where $\alpha$ and $\beta$ are two homomorphisms from these onto the same finite group $F$ of order $f$.

(Here the $2$ in $2L$ and $2R$ refers to quaternion groups and the fact that the quaternions form a double cover; it's not meant to mean that these groups are e.g. $2.L$). Just as the notation suggests, the order of $\frac1f[\![L\times R]\!]$ is $\dfrac{\left|L\right|\cdot \left|R\right|}{f}$).

EDIT: Note that unlike what the comments (and a previous version of this answer) suggested, this is not just a quotient of the product group by an element $[\![i_L,i_R]\!]$, where $i_L$ and $i_R$ are elements of order two in the respective groups. (This would pick out e.g. all the elements where $l$ is not a reflection.) Instead, it's in some sense the opposite; it consists of all pairs $[\![l,r]\!]$ where $l$ and $r$ have the same 'parity' — for instance, in the case where $L=R=D_{2p}$, those pairs where either both $l$ and $r$ are reflections or where neither is. It's not too hard to see that this is a subgroup, since the property that $l$ and $r$ both have the same parity (more generically, are in the same homomorphism class) is preserved under group multiplication.

The notation $(p\times p):2$ appears in chapter 16 of The Symmetries of Things; it refers to the split extension of the group $H=p\times p$ (i.e. $C_p\times C_p$) by a group $2$ ($C_2$), with the nontrivial element $c$ of $2$ acting on elements $h\in H$ of the other group by $c^{-1}hc=h^{-1}$. This can naturally be thought of as the dihedralization of $p\times p$; of course the dihedralization of $C_p$, i.e. $p:2$, is just $D_{2p}$.