12 votes
Accepted

Distinct characters with the same character values, outer automorphisms and Galois conjugation

Take $G = S_3 \times S_4$ and consider the unique two-dimensional irreducible representation of $S_3$ and the unique two-dimensional irreducible representation of $S_4$. These have the same character ...
Will Sawin's user avatar
  • 133k
10 votes

Regular orbits for automorphisms of finite simple groups

By a result of Horoševskiĭ you can never find such an automorphism, that is all automorphisms of finite simple groups have a regular orbit.
Michael Giudici's user avatar
10 votes
Accepted

Nonisomorphic central products on the same pair of groups?

The smallest example: $G = H = \mathbb{Z}/4 \times \mathbb{Z}/2$, generated by say $x$ of order 4 and $y$ of order $2$, and $A = B = \langle x^2, y \rangle \cong \mathbb{Z} / 2 \times \mathbb{Z} / 2$. ...
9 votes

Regular orbits for automorphisms of finite simple groups

As pointed out by Michael Giudici the answer is given by a result of Horoševskiĭ. Here is a proof following the paper by Horoševskiĭ. Lemma: Let $\phi$ be an automorphism of $G$ with $|\phi|$ ...
testaccount's user avatar
9 votes
Accepted

What are double groups mathematically?

As far as I can tell, a double group is a double cover of a group. Specifically, if $G \subset \operatorname{SO}(n)$ is a group acting by rotations of $n$-dimensional space, its double group is the ...
Theo Johnson-Freyd's user avatar
7 votes
Accepted

Classification of non-abelian simple groups with cyclic T.I. Sylow p -subgroup

This is answered in the paper of Harvey Blau, "On Trivial Intersection of Cyclic Sylow Subgroups" Proc AMS 1985. Whenever a Sylow $p$-subgroup of a finite simple group is cyclic, it is T.I. ...
Dave Benson's user avatar
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5 votes
Accepted

Distribution of 2-groups

The following is an empirical argument to show that the total number of (isomorphism classes of) groups of order less than $2^m$ is dwarfed by the number of order exactly $2^m$. The number of groups ...
Dave Benson's user avatar
  • 11.2k
4 votes

Regular orbits for automorphisms of finite simple groups

If I am reading your question correctly, then I think $A_{5}$ is an example where this fails. The automorphism group is isomorphic to $S_{5}$. The only elements of composite order in the automorphism ...
Geoff Robinson's user avatar
4 votes

The sum (with multiplicity) of the cubes of irreducible character degrees of a finite group

It turns out that my original question does indeed have a positive answer. In fact, one can show that if $G$ has an irreducible character of degree $\geq 3$ then ${\rm AD}(G) \geq 2+ |G'|^{-1}$. The ...
1 vote
Accepted

An explicit matrix form

It looks like you are working with respect to the orthogonal form with matrix $\begin{pmatrix} & w_0 \\ w_0 \end{pmatrix}$, where $w_0 = \operatorname{antidiag}(1, \dotsc, 1)$. That's the one ...
LSpice's user avatar
  • 11.1k

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