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.
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When are subgroups of a 2-Generated group also 2-Generated
In this question it is asked whether subgroups of a finite, 2-generated group are also 2-generated. The answer is no, with a nice counterexample. However, when the group is Abelian, this is true. My ...
- 3,044
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Inclusions of product of groups
I have the following group-theoretic question, where I don't know what really could/should happen:
I have three groups, $G_1 \cong \mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/m\mathbb{Z}$, $G_2 \cong \...
- 769
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Matching the orders of semidirect products
Group $C:=A:B$ where ord($A$)= $2^7$ and ord ($B$)= $72$. There is a group $D$ such that $D\cap A$ has order $32$ and $D\cap B$ has order $18$. It is given that $C\cap D$ has order $2304$. But the ...
- 1,291
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Writing symmetry operations as composition of group action and stabilizer
In the attached image let $K$ be the union of all colored-in edges. The symmetry group of the cube is restricted by imposing that K should be mapped onto itself. I want to find the restricted symmetry ...
- 73
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Proof that the symmetric groups of two sets with equal cardinality are isomorphic to each other? [closed]
I wrote a "proof" of the following theorem, and would like to know if it's correct, or what went wrong and so on. Any feedback is appreciated. Thank you in advance.
Let $S$ and $T$ be sets, ...
- 57
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Bruhat Order of the Finite Symmetric Group
I am studying Theorem 2.1.5 in "Combinatorics of Coxeter Groups", but I am confused by a statement in the proof of the "if" direction (the part after $\textbf{However}$). Let me ...
- 63
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A basic question about supersolvable quotients
If $G$ is a finite group and $N_i\trianglelefteq G$ ($i=1,2$) such that $G/N_i$ is supersolvable, then, is it true that $G/(N_1\cap N_2)$ is supersolvable?
One may give only hint; I will try to prove ...
- 3,043
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Conjugacy classes of normal subgroup in group
Let $G$ be a finite group, and $N$ be a normal subgroup of index $p$.
If every conjugacy class of $N$ is also a conjugacy class in $G$, what can we say about $G$ or $N$?
Such instances occur if $G$ is ...
- 3,043