# Questions tagged [group-theory]

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory is the study of groups.

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### inverse of isomorphism from $\mathbb{C}[G]$ to $\bigoplus_V \text{End}(V)$

I understand that for any finite group $G$, there is an isomorphism from the group algebra $\mathbb{C}[G]$ to the direct sum of endomorphisms in which each irreducible representation of $G$ appears ...
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### How can I learn about the Monster group?

There are several questions about the Monster group on this site, but none really answer the question in the title. While reading about groups in a first year algebra course, I was told about the ...
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### Normal subgroups orbits

Let $G$ be a topological group acting transitively and effectively on the space $X$ and let $J,K$ be two normal subgroups of $G$ such that $G=J\cdot K$ and $J\cap K\not =\{e\}$. Let $Gx_0$ be the ...
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### Maximal order of a torsion element in a hyperbolic group

Suppose that $G = \langle X \rangle$ is a $\delta$-hyperbolic group, i.e. all geodesic triangles are $\delta$-thin (the inverse image of a point under the projections onto a tripod has diameter ...
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### Assistance with Wielandt's permutation group problem involving blocks containing pairs

EDITED to clarify and incorporate the comments and improve notation (sorry the original was poor). Thank you Derek Holt and Mesel. Let $G$ be a finite group acting on a set $\Omega$. A block is a ...
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### Could every finite simple group be related to a pair of Lie Groups?

In terms of the classification of simple groups, it is known that every finite simple group is either: Cyclic Alternating Of Lie type One of the 26 Sporadic groups On the other hand there are 5 ...
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### Explanation of a proof of the Second Sylow Theorem (Conjugation of Sylow p-subgroups)

I am an undergrad Mathematics student and I've been reading some additional literature for my lectures and came upon a quite short and seemingly elegant proof of the Second Sylow Theorem. Though, I ...
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### Determination of the unitary irreps of $\mathbb{R}$ using Stone's theorem

I've tried to find the unitary irreducible representations of the additive group $(\mathbb{R},+)$ and came up with a pair of results, which I want to verify if are correct. They are: Theorem: Let the ...
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### Complete reducibility for the Poincaré group

If $(\rho,V)$ is a unitary representation of a group $G$ which is finite dimensional, then complete reducibility is kind of easy to prove. Indeed, if $V$ is not irreducible, then it has one proper ...
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### Isomorphism preserving operations

Questions : What are group modification processes ( group transformations ) which preserve the isomorphism property. One operation is abelianization of group does not preserve isomorphism. I mean to ...
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### What are the topological groups the points of which are separated by their finite-dimensional representations?

It is known that the continuous finite-dimensional irreducible unitary representations separate the points of Hausdorff compact topological groups (this is, in part, the Peter-Weyl theorem). These, ...
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### Geometry and equivalence of orbit spaces

Consider the action of $G = {\rm SO}(2)$ by conjugation on real symmetric matrices, that is: $$g\cdot A = gAg^{-1} = gAg^{t},$$ for $A$ a symmetric real matrix $2 \times 2$. The author says that I can ...
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### When Burnside's Lemma does not apply

I have a finite group G which acts on a set X. I want to establish the number of distinct members of a subset Y $\subset$ X, however Y is not closed under G (I think that the correct terminology here ...
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### Order of $x$ in $\mathbb F_q[x]/(g(x))$

Let $\mathbb F_q$ be a finite field and $g(x)$ be a polynomial in $\mathbb F_q[x]$ with non zero constant term. Given the factorization of $g(x)=p_1(x)^{e_1}\dots p_k(x)^{e_k}$, give an estimate for ...
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### Who discovered that normalizer of an abelian Sylow $p$-subgroup controls $p$-transfer?

Theorem: Let $G$ be a finite group and let $P\in Syl_p(G)$ and assume $P$ is abelian. Then $N_G(P)$ controls $p$-transfer. I wonder who discovered above theorem? Is it due to Burnside?
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### Find all automorphisms of $\mathbb{Z}_{10}$

Since $\mathbb{Z}_{10}$ is a cyclic group (ex. generated by $\left<1\right>$), all automorphisms will be determined by finding $\phi(1)$. Since we want an isomorphism, we map $1$ to a generator, ...
### Proof verification that Aut$(A_4) \simeq S_4$
I've been watching a video proving $S_4 \simeq$ Aut$(A_4)$, but I was having trouble making sense of it. I wasn't quite sure I even understood exactly what I didn't understand, so I set out to prove ...
### Action of $S_3$ on a set of subsets
Consider the group $S_3=\{1,x,x^2,y,xy,x^2y\}$ where $x=(123),\ y=(12)$. This group acts on the set $\mathcal P_3(S_3)$ of subsets of $S_3$ of cardinality 3 by left multiplication. Describe the ...