# Tagged Questions

Questions on group theory which concern finite groups.

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### Does the Fano plane “embed” in the complex projective plane?

$PSL(2,7)$ acts on the projective plane over $\mathbb{F}_2$ (the Fano plane) through its identification with $GL(3,2)$. It also acts on the projective plane over $\mathbb{C}$ through either of its ...
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### On the structure of groups according to their conjugacy classes

A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n-1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...
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### Even-odd partitioned groups! [closed]

Let $G$ be a group with the property $G=G_e\dot{\cup} G_o$ with $G_oG_o\subseteq G_e\leq G$. ($\dot{\cup}$ denotes disjoint union, $\leq$ is subgroup notation, and $G_o^{-1}=\{x^{-1}: x\in G_o\}$) ...
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### Uniqueness of the fusion ring for simple finite group

We know that the irreducible representations $R_i$ of a group $G$ can give rise to a fusion ring: $R_i\otimes R_j = \oplus_k N^{ij}_k R_k$. I wonder if the following statement is true or not: If $G$ ...
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### Finite groups: equations with many solutions

Let $\omega$ be a word in the free group on generators $x_1,x_2,\ldots,x_n,g_1,g_2,\ldots,g_k$, where $n>0$ and $k\geq 0$. For any finite group $G$ and elements $g_1,g_2,\ldots g_k$ in $G$ we ...
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### Must normalizing field outer automorphisms “divide” the dimension?

Imprecise question: To get a normalizing field outer automorphism of order $r$, must we multiply the dimension by $r$? Precise hypothesis: Let $p\geqslant 5$ be a prime, let $q$ be a power of $p$ and ...
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### The parity of the full automorphism group order of finite non-abelian groups of prime exponent

Is there a finite non-abelian group $G$ of prime exponent such that the full automorphism group of $G$ is of odd order?
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### For a new operation on a finite group of odd order giving a loop structure, when does this also gives a group

For finite groups $G$ of odd order, as $x \mapsto x^2$ is bijection (but no automorphism in general) then, we can define for each $g \in G$ the element $x^{1/2}$ by requiring $(x^{1/2})^2 = x$. Then ...
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### What kind of group invariants exist? [closed]

Let $G$ be a finite group.Then it is known, that: 1) The group determinant determines the group (up to isomorphism) 2) The 1, 2, 3 characters determine the group 3) The invariants $f_1,\cdots,f_m$ ...
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### Perfect group of order 190080

I need to know some properties of the perfect group of order $190080$ which is the Schur cover of the Mathieu group ${\rm M}_{12}$, but when using ...
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### embedding of $O_4^-(q)$ in $U_4(q)$

For $q$ an odd prime power, one can construct $H:=O_4^-(q)<U_4(q)$ by treating the $H$-invariant bilinear form as Hermitean. On the other hand $U_4(q)$ is isomorphic to $O_6^-(q)$; I need to ...
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### Frobenius Groups of Automorphisms

Recently, I am looking different papers on the topic $$\mbox{Frobenius groups of automorphisms of a group.}$$ But I am familiar with Frobenius groups only; not with their (faithful) actions on groups. ...
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### Connection between cyclic group and exponential function

I have been thinking about this for a while, but now got to the point where I got stuck. I don't know if it might be considered as a research level question, but I would be very happy if somebody knew ...
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### Restriction of $H^3(M_{24}, U(1))$ to $M_{12} \rtimes \mathbb{Z}_2$

$M_{12}\rtimes \mathbb{Z}_2$ is a maximal subgroup of $M_{24}$, where $M_{24}$ and $M_{12}$ are Mathieu groups . Also, it is known that $H^3(M_{24}, U(1)) \cong \mathbb{Z}_{12}$. I want to find the ...