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8h

comment 
Computing a transversal of a subgroup $H$ of $G$ in expected $O(G : H^2 \log G : H + H)$ time
It may be that this question will come to Derek Holt's notice in due course. He should have more insight than most into the issues you raise. 
2d

revised 
real representation of a product group
typo 
2d

revised 
real representation of a product group
added "associative" 
May 26 
comment 
Finite groups whose nontrivial elements have no fixed points
Probably better on Mathstackexchange. 
May 24 
comment 
Localized at $p$ integral representations of finite elementary $p$groups
That is the cyclic case. I the noncyclic elementary Abelia case, (which you have to look further back for, working from the references in Jones' paper), there are infinitely many isomorphism classes when $ n >1.$ 
May 23 
revised 
A simple group that its order divide order of an alternating group
explained original question changed. 
May 23 
revised 
For $k>3$ does there exist an odd prime $q_k$ such that $p_k=2^kq_k+1$ is prime and $p_k$ divides $a_k=\dfrac{3^{2^{k1}}+1}{2}$?
typo 
May 23 
revised 
A simple group that its order divide order of an alternating group
typo 
May 23 
answered  A simple group that its order divide order of an alternating group 
May 23 
comment 
Localized at $p$ integral representations of finite elementary $p$groups
There are infinitely many isomorphism types for $n >1.$ The number of isomorphism types is actually at least $1 + 2h_{p}$ when $n =1$, where $h_{p}$ is the class number of $\mathbb{Z}[\omega]$ and $\omega$ is a primitive $p$th root of unity. For references, see for example projecteuclid.org/euclid.mmj/1028998908. 
May 22 
comment 
For natural numbers 1 to n, is the square of their sum equal to the sum of their cubes?
This probably belongs on Mathstackexchange. For any positive integer $k$, the sums $\sum_{j=1}^{n}j^{k}$ are wellstudied and reasonably well understood. In particular, it is wellknow and easy to prove by induction that $\sum_{j=1}^{n}j = \frac{n(n+1)}{2}$ and $\sum_{j=1}^{n} j^{3} = \left(\frac{n(n+1)}{2}\right)^{2}.$ This explains your observations and verifies that the answer is positive for all $n$. 
May 22 
revised 
For $k>3$ does there exist an odd prime $q_k$ such that $p_k=2^kq_k+1$ is prime and $p_k$ divides $a_k=\dfrac{3^{2^{k1}}+1}{2}$?
expanded explanatio 
May 22 
revised 
real representation of a product group
added extra sentence of explanation 
May 22 
revised 
real representation of a product group
added alternative viewpoint 
May 22 
answered  For $k>3$ does there exist an odd prime $q_k$ such that $p_k=2^kq_k+1$ is prime and $p_k$ divides $a_k=\dfrac{3^{2^{k1}}+1}{2}$? 
May 22 
revised 
real representation of a product group
typospacing issue 
May 20 
revised 
real representation of a product group
expanded 
May 19 
comment 
A Diophantine equation with prime powers
In the Pell like case, how do you decide whether $x$ is prime or not, and if so, whether $y$ is also prime? 
May 19 
revised 
real representation of a product group
gave more explanation 
May 19 
awarded  Enlightened 