8 votes

What exactly is the orbit-stabilizer theorem?

The way I state it in my final year group theory course is: Let $G$ act on $\Omega$ and let $\alpha \in \Omega$. Then there is a bijection between the right cosets $G_\alpha g$ of $G_\alpha$ in $G$ ...
Derek Holt's user avatar
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6 votes
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Triple-Transitivity/"Specify three know all" property of exotic transitive $S_5\subset S_6$

The group you call the exotic $S_5$ is otherwise (and better) known as ${\rm PGL}(2,5)$, and the three properties you described are collectively known as sharp triple transitivity. The group ${\rm PSL}...
Derek Holt's user avatar
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6 votes

Smallest group acting transitively on projective space

Over finite fields you get much more. For example (images of) the Singer cycle (and overgroups). As a somewhat random example, there are (up to conjugacy) 7 minimal transitive subgroups of $PGL(4,3)$, ...
ahulpke's user avatar
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4 votes

On the Normality of Subgroups in Finite Groups

Consider cases where $G=N\times M_1$ and $M<M_1$. Then we trivially have $N\triangleleft G$ and $N\cap M=1$, and things in $M,N$ commute with one another so $[M,N]\leq\textrm{anything}$. But if $M$ ...
Gareth McCaughan's user avatar
4 votes
Accepted

If the conjugacy class is preserved under product.

Counterexample in $S_4$: $a_1=(12), a_2=(13)$ and $b_1=(1234), b_2=(1432)$: $a_1b_1=(234), a_2b_2=(14)(23)$, and these elements have different cycle type hence not conjugate.
Nicky Hekster's user avatar
4 votes

Smallest group acting transitively on projective space

For $K = \Bbb R$ it follows from Montgomery & Samelson, "Transformation Groups of Spheres" that one can do better than $\operatorname{SO}(n)$ when $n \equiv 0 \pmod 2$ (and $n > 2$) ...
Travis Willse's user avatar
3 votes
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On the finite minimal non-solvable groups

A natural place to look for examples would be to look for a minimal simple group with non-cyclic Schur multiplier. The Schur multiplier of the Suzuki group $S = {}^2B_2(8)$ is elementary abelian of ...
testaccount's user avatar
3 votes
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$\varphi^G$=$\psi^G$ implies that subgroups are conjugate

There appears to be some literature around the study of such pairs of subgroups, either just named "subgroups inducing the same permutation representation" like in this article titled, well, ...
Bruno B's user avatar
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1 vote
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About the two step centralizer

For $p$-groups of maximal class, one reference is Leedham-Green and McKay's The Structure of Groups of Prime Power Order: Proposition 3.1.4. Let $G$ be a finite $p$-group of order $p^n$ and of ...
Arturo Magidin's user avatar

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