group theory notation (Conway-Hulpke-McKay, On Transitive Permutation Groups)

Question

I've been reading the paper "On Transitive Permutation Groups" by Conway-Hulpke-McKay, and I am having some difficulty deciphering some basic notations. Take for example the groups of order 8:

• the second group there is $4 [\times] 2$ --- I assume that's the direct product $Z/4 \times Z/2$?

• the fourth group is $D_8(8) = [4]2$ --- I assume that's a non-trivial semi-direct product $Z/4 \rtimes Z/2$? BUT:

• the 7th group is $\frac{1}{2}[2^3]4$ --- this is a subdirect product of $[2^3]4$, so the latter is not a non-trivial semidirect product?

I know that these are standard notations, but I could not find a reference that explain what they are. Many thanks for your help.

Answer 1

This is a notation (which originated with John Conway) that indicates not just the isomorphism type, but the permutation action. The brackets (as in $[\times]$) indicate that this product construction is not just abstract, but is seen in the permutation structure.

Conway liked points 0..7, but the computer cannot do this, thus 8 stands for 0 in the following. You should imagine the points arranged as

8 1 2 3
4 5 6 7

or

8 4
1 5
2 6
3 7

with blocks of imprimitivity being vertical.

Now the direct product $4[\times]2$ has (first diagram) the element of order 4 acting horizontally $(8,1,2,3)(4,5,6,7)$, and the one of order 2 acting vertically $(8,4)(1,5)(2,6)(3,7)$.

In the case of $[4]2$ (second diagram) we have the element of order $4$ acting vertically, and the element of order 2 permuting the blocks, but with a twist that gives the action so the product is not direct. The fact that the name is not $4[\times]2$ indicates this nontrivial action.

Finally,the $\frac 12$ in front indicates a construction that takes an existing group (with nice generators) and builds from these a new group (not necessarily a subgroup!) that has half the order. (This makes an implicit assumption that there are not too many groups of this kind so that the name determines the transitive group as unique.)

Take $H$ the TransitiveGroup(8,20) which has order 32 and name $[2^3]4$. This indicates that there is a block system with 4 blocks of size 2, with an action on the blocks the transitive cyclic group on 4 points. Let $K$ be the kernel of that action (which has order $32/4=8$) It has (the blocks) 4 orbits of size 2, but it is not the full direct product $2^4$, but only of order $2^3$. It is generated by $b=(8,1,2,3)(4,5,6,7)$ and the element $a=(2,6)(3,7)$.

This all lies in a group $[2^4]4$ in which the 2's act vertically and the 4 horizontally, which is generated by $(1,5)$ and $b$.

If we now multiply both generators of this wreath product with $(3,7)$, we get $a'=(1,5)(3,7)$ and $b'=(8,1,2,3,4,5,6,7)$. (This is not a subdirect product, maybe a "subwreath" product) These generate a group that has only half the order of $H$ and is the group $\frac 12[2^3]4$.

Overall this is an ad-hoc scheme that manages to give descriptive names up to degree 15. (Doing it for degree 12 already was a slight struggle.) It would break down attempting it for the many more groups of degree 16.