Questions tagged [group-theory]
For questions about the study of algebraic structures consisting of a set of elements together with a well-defined binary operation that satisfies three conditions: associativity, identity and invertibility.
51,422 questions
4
votes
1
answer
107
views
Bounding indexes of normal cores, when there is extra information
Let $H\leq N\leq G$ be groups with finite indexes in each other, and with $N$ normal in $G$. An exercise often given to students is to show that ${\rm core}_G(H)$ (the largest normal subgroup of $G$ ...
- 747
1
vote
0
answers
35
views
Connections between roots of unity's automorphisms and general polynomial solvability by radicals
Is there a genuine relation between the automorphisms that act upon complex roots of unity, and the solvability by radicals of the same degree?
To clarify what I mean, I'll explain what I noted.
...
- 29
2
votes
0
answers
47
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Mathieu groups via Steiner systems
Can one suggest an elementary reference for Mathieu groups via Steiner systems?
The one I am following is the book on Permutation groups by Dixon et al.
The other reference I am looking at is the one ...
- 3,643
1
vote
0
answers
34
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is extension of an a-T-menable group by an amenable group still a-T-menable? [closed]
let $1 \rightarrow A \ \rightarrow G \rightarrow Q \rightarrow 1$ be a group extension, suppose $A$ is amenable, and $Q$ is a-T-menable (An a-T-menable group is a topological group that admits a ...
- 117
8
votes
1
answer
213
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Prove Rubik's Cube group is isomorphic to $(\mathbb Z_3^7 \times \mathbb Z_2^{11}) \rtimes \,((A_8 \times A_{12}) \rtimes \mathbb Z_2)$
On the Wikipedia page for the Rubik's Cube group $G$, it states without citation that
$$
G \cong(\mathbb Z_3^7 \times \mathbb Z_2^{11}) \rtimes \,((A_8 \times A_{12}) \rtimes \mathbb Z_2)
$$
And ...
- 7,864
5
votes
1
answer
61
views
Simple groups containing $S_n$ or $A_n$ as an index $p$ subgroup
It's well known that $PSL(2,7)$ is a simple group containing $S_4$ as an index $7$ subgroup, and $PSL(2,11)$ is a simple group containing $A_5$ as an index $11$ subgroup.
My question is, does ...
- 2,882
1
vote
1
answer
49
views
regular tiles on a sphere
Is there a similar idea to the wallpaper groups, but on a sphere? Vertical strips seems to be a group.
I can also divide the surface into 8 pieces with one cut along 360 degrees of meridian and one ...
- 416
2
votes
0
answers
39
views
Fields of values of extensions of irreducible characters.
Let $G=\langle N,x\rangle$ be a finite group, where $N$ is a normal subgroup of $G$ and $x$ is an element of $G$. So $G$ is generated by $N$ and $x$, and $G/N$ is a cyclic group. Let $t$ be the order ...
- 21
4
votes
0
answers
121
views
If $K$ is a normal subgroup of $G$, then $K'$ (the commutator subgroup of $K$) is a normal subgroup of $G$; alternative proof to the standard one?
If $K$ is a normal subgroup of $G$, then $K'$ (the commutator subgroup of $K$) is a normal subgroup of $G$.
I know $K'\text{ char }K$ and if $H\text{ char }K\triangleleft G$, then $H\triangleleft G$, ...
- 81
4
votes
1
answer
217
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What is an orbit space?
I am unsure about the definition of the orbit space. Take a covering space $E$ and consider the group action $\Gamma$ of homeomorphisms acting on $E$. Then $E/\Gamma$ is an orbit space.
Let $X = S^1$ ...
- 3,143
2
votes
0
answers
42
views
Can representations induced from different subgroups of index 2 be equivalent?
Let $G$ be a group and $H_1, H_2$ be two distinct subgroups of index $2$. Let $\chi_1, \chi_2$ be two $1$-dimensional representations of $H_1$ and $H_2$ respectively (over an algebraically closed ...
- 540
4
votes
1
answer
86
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Are the additive groups $\ell^\infty(\omega;\mathbb{Z})$ and $\ell^\infty(\omega;\mathbb{Z})^\omega$ isomorphic?
Let $\ell^\infty(\omega; \mathbb{Z})$ be the additive group of uniformly bounded sequences $\omega \to \mathbb{Z}$ with pointwise addition. Is it isomorphic to the (countably) infinite direct product ...
- 26.2k
2
votes
1
answer
109
views
Diagram chasing mod $S$
$$\require{AMScd}
\begin{CD}
A @>{\phi}>> B @>{\psi}>> C\\
@V{\alpha}VV @V{\beta}VV @V{\gamma}VV\\
A^\prime @>{\mathsf{id}}>> B^\prime @>{x \mapsto xA^\prime}>> {B^\...
- 103
6
votes
1
answer
124
views
Given a finite group $G$, what is the smallest magma $M$ satisfying $\operatorname{Aut}(M) \cong G$, in particular for $G=D_5,A_5$?
From the answer to this question we know that every finite group $G$ is the automorphism group of some finite semigroup $S$, but the construction doesn't give any relationship between the orders of $G$...
- 71
0
votes
1
answer
91
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Normal subgroup of a finite index subgroup has a finite index subgroup normal in ambient group? [closed]
Let $G$ be a group. Let $H$ be a subgroup of $G$ with finite index $[G:H]$. Let $N$ be a normal subgroup of $H$. Is there a subgroup $N'$ of $N$, such that $N'$ is normal in $G$, and $[N:N']$ is ...
- 1,640