# Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

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### Real non-principal 2-blocks for finite groups of Lie type

Is it true that each finite group of Lie type has a non-principal real $2$-block? Here the principal block is the one containing the trivial character and a block is real if it contains the complex ...
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### What is the normal subgroup generated by nonsingular, positive definite matrices? [closed]

Within $GL(n,\mathbb C)$ let $H$ denote the normal subgroup generated by products of positive-definite, Hermitian matrices. What is this subgroup? And what is the quotient $GL(n,\mathbb C)/H$? Do ...
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### Large gaps in the norm of a subgroup and its centraliser

Take an infinite finitely generated group $G$ with an infinite subgroup $N$ which has an infinite centraliser $Z = Z_G(N)$. Let $S$ be some [symmetric] generating set of $G$ and for $g \in G$, ...
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### Minimal generation of simple groups and Ore's conjecture

The well known Ore's conjecture (now established) states that every element of a finite non-abelian simple group $G$ is a commutator of a pair of elements. Also we know that $G$ is $2$-generated. I ...
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### Image of the mapping class group of surfaces into automorphism group?

Let $S_{g,p}^n$ be a compact oriented surface of genus $g$ with $p$ punctures and $n$ boundary components, and $\operatorname{Mod}(S)$ and $\operatorname{PMod}(S)$ be the mapping class group and the ...
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### $G/F(G)$ is isomorphic to $X_1\times\cdots\times X_t$

I asked this question on math.stackexchange many hours ago, but haven’t got an answer. It was mentioned in a comment that the answer to my question is trivial, but I couldn’t see why. $G$ is a finite ...
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### Is the solvable radical of a finite perfect group contained in the Schur multiplier of the quotient of the group modulo the solvable radical?

Let $G$ be a finite perfect group, and let $N$ be the solvable radical of $G$. If $G/N$ is a non-abelian simple group, then is it true that $N$ is contained in the Schur multiplier of $G/N$? If this ...
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### QI-closure of $\mathrm{NA}\times\mathrm{NA}$

Suppose we know the following about a class of groups $\mathcal{G}$. If $G$, $H$ are f.g. r.p. nonamenable groups, then $G \times H \in \mathcal{G}$. If $G \in \mathcal{G}$, $G$ is f.p., and $G$ is ...
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### Finite group ${\rm Sp}_4({\Bbb F}_3)$: involutions coming from a 4-dimensional complex representation

I am interested in the finite unitary reflection group $G= G_{32}$, the group No. 32 in Table VII on page 301 of the paper: Shephard, G. C.; Todd, J., A. Finite unitary reflection groups. Canad. J. ...
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### Properties of the spectrum of the Koopman representation

Let $G$ be a discrete countable infinite group acting on a compact metric space $X$ via homeomorphisms preserving a probability measure $\mu$. A function $\lambda\colon G\to \mathbb C$ is an ...
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### Are there perfect DTI-groups which are not simple?

Recall that a subgroup $H$ of a group $G$ is called a TI-subgroup if for every $g \in G$ it is $H \cap H^g \in \{1,H\}$. We say that a group $G$ is a DTI-group if the derived subgroups of all of its ...
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### What can we get from an automorphism of order $2$

An automorphism of order $2$ is an automorphism fixes some elements and inverts the others. It’s well known that not all groups have automorphisms of order $2$, $C_2$, for example. But if a group ...
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