# On group-theoretic shorthand notation

I have often seen shorthand notation used in group-theoretic contexts and I believe it is called ATLAS notation. However, even with some searching I have not been able to find a satisfactory summary of the meaning of this notation.

I have found the meaning of some things, like $A \times B$ for the direct product of $A$ and $B$, $G=A.B$ for a group with a normal subgroup isomorphic to $A$ and for which $G/A \cong B$, $A:B$ when $A.B$ is a split extension, and $A \cdot B$ when $A.B$ is not split.

However, I have seen things like $4 \times 4 \times 4$ and $3^3.\mathrm{SL}_3(3)$. What do numbers by themselves, possibly with powers, represent - the cyclic groups of those orders, perhaps?

What is a good, complete reference for this notation, available freely online? If no such thing exists, a nice explanation in an answer would also work.

Thanks.

EDIT: Here is an example of a paper where I have seen this notation.

• Could you give us an example of a source which uses this notation? – Michael Albanese Aug 9 '14 at 14:23
• @DietrichBurde Yes, I do know that. I was using a real example I have seen but asking about the $3^3$ part, as indicated in the next sentence. – Alex Petzke Aug 9 '14 at 14:35
• The Wiki page on finite simple groups has $A_2(3^3)$, where $3^3=27$. What is your source for the $3^3$ notation ? – Dietrich Burde Aug 9 '14 at 14:38
• If it is ATLAS notation, you probably want to look in the ATLAS of finite simple groups. Your library probably has a copy. – user1729 Aug 9 '14 at 15:47
• There is a web version of the ATLAS, but I don't see any explanation there for ATLAS notation, which appears on Page xx of the printed version. Your guess is correct that integers represent cyclic groups of the indicated order. Expressions of the form $p^n$ denote elementary abelian groups, but more complicated expressions such as $p^{a+b}$ indicate more structure, and $p^{1+2n}$ is an extraspecial $p$-group, for instance. – James Aug 9 '14 at 20:37