Simple group with given order I would like to know: Is there any simple group of order $2^{n}(2^{n}-2)$? 
This is a special case of a group of order $p^{2}-1$ when $p$ is Mersenne prime. Thanks.
 A: For a simple numerical invariant, one can use that a known, non-abelian, finite, simple group $G$ always satisfies $|U|^2 < |G|$ when $U$ is a Sylow $p$-subgroup of $G$. This was shown in (Kimmerle, et al., 1990, Theorem 3.6) and (Mazurov–Zenkov, 1995, Corollary 1).
However, if $p=2^n-1$ is prime, $|G|=p^2-1 = 2^n(2^n-2) = 2^{n+1}(2^{n-1}-1)$,and $U$ is a Sylow 2-subgroup, then $|U|^2 = 2^{2n+2} > 2^{2n} > |G|$.
Proof sketch
Here is an intuitive argument why a Sylow $p$-subgroup $U$ should satisfy $|U|^2 \leq |G|$ when $O_p(G)=1$: Consider $|U|^2 = |U| |U^g| = |U U^g | |U \cap U^g| \leq |G| |U \cap U^g|$. If $|U|^2 > |G|$, then $|U \cap U^g | > 1$. In most groups, $O_p(G) = U \cap U^g$ for some $g$, so we'd have a non-trivial $p$-core. Of course in some groups it requires 3 Sylow $p$-subgroups to get the $p$-core, so this is not a proof in general. However, (Mazurov–Zenkov, 1995) uses induction and a result on defect groups of blocks (and a small amount of case by case analysis) to show that in a known, non-abelian, finite, simple group, the $p$-core is always the intersection of two Sylow $p$-subgroups.
Bibliography


*

*Kimmerle, Wolfgang; Lyons, Richard; Sandling, Robert; Teague, David N.
“Composition factors from the group ring and Artin's theorem on orders of simple groups.”
Proc. London Math. Soc. (3) 60 (1990), no. 1, 89–122.
MR1023806
DOI:10.1112/plms/s3-60.1.89

*Mazurov, V. D.; Zenkov, V. I.
“Intersections of Sylow subgroups in finite groups.”
The atlas of finite groups: ten years on (Birmingham, 1995), 191–197, 
London Math. Soc. Lecture Note Ser., 249, Cambridge Univ. Press, Cambridge, 1998.
MR1647422
DOI:10.1017/CBO9780511565830.019
A: You should've thought of just going through the list and see if you could find one with your desired order! Here it is : 
http://en.wikipedia.org/wiki/List_of_finite_simple_groups
Unless you didn't know that finite simple groups were already fully classified up to isomorphism. =)
But since you speak about Mersenne Primes, don't bother about going through the list if you want to assume that $2^{n-1} - 1$ is prime, because Burnside's theorem says that you won't find any such group (Seirios did the comment first, but I must admit any group theorist who has once seen Burnside's theorem remembers it for the rest of his life everytime someone speaks of the order of a group! It is a must-see if you haven't yet.)
Hope that helps!
