Tagged Questions

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 studies groups.

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Finding a 3-embedded subgroup.

I have the group of order 108 $G=(((\mathbb{Z}_3 \times \mathbb{Z}_3)\ltimes \mathbb{Z}_3)\ltimes \mathbb{Z}_2) \ltimes \mathbb{Z}_2$ obtained from an algorithm in GAP, but I need to prove that it has ...
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Product with a normal subgroup

If $H \unlhd K \unlhd G$ and $P\unlhd G$ then does necessarily $HP \unlhd KP$? I can see this is true using the correspondence theorem since $HP/P \unlhd KP/P$ I want to try direct and prove it ...
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Commensurability of two groups

If two groups $G\bigoplus \mathbb{Z}$ and $H\bigoplus \mathbb{Z}$ are commensurable. Does it imply that the groups $G$ and $H$ are commensuarble?
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Supersolvable and pronormal subgroups

Let $G$ be a finite group such that all subgroups of prime-power order are pronormal in $G$. If $M$ is a normal $p$-subgroup of $G$ then all prime-power order subgroups of $G/M$ are pronormal in $G/M$....
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kernel of product of group homomorphisms

Let $f,g:A \to B$ be group homomorphisms, with $B$ abelian. Then $f\cdot g$ is also a group homomorphism. What can I say about $\ker(f \cdot g)$ in terms of $\ker(f)$ and $\ker(g)$?
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Existence of surjective map / homomorphism from an infinite group onto its symmetric group

From Cayley's theorem , we know that any group $G$ can be embedded in its permutation group $S(G)$ ; I would like to ask , If $G$ be an infinite group , then does there exist a surjection from $G$ ...
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Lie algebra equivalent definition

I am reading the paper "Affine projections of polynomials" by Neeraj Kayal. I need a clarification regarding the equivalence of two definitions of lie algebra : Let $f\in\mathbb{F}[x_1,\ldots,x_n]$ ...
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Cohomology of a group of order two with coefficients in a finite abelian group of odd order

I am looking for an elementary proof that the cohomology groups in the title are trivial in the positive degrees. In more detain, let $G=\{1,s\}$ be a group of order two, and let $A$ be an abelian ...
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A block in understadning a proof of Sylow's Theorem [duplicate]

If $G$ is a finite group of order $n$ and $p$ is a prime divisor $|G|$, then according to one of Sylow's theorems $G$ has a subgroup of order $p^m$, where $p^m$ is the highest power of $p$ which is a ...
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Group action with two normal subgroups which induce same block system

So awhile back I asked this question here on stack exchange: Normal subgroup $H$ of $G$ with same orbits of action on $X$. At the time I wasn't quite sure what I was really wanting to know about ...
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Simple characterization of integers among abelian groups

This is part of an early exercise in Freyd's abelian categories. Let $\mathscr{G}$ be the category of abelian groups. The group of integers is distinguished, up to isomorphism, by the facts that: ...
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If $G$ a finite $p$-group s.t $G/[G;G]$ cyclic then $G$ abelian.

Let $G$ be a finite $p$-group and $[G,G]$ its commutator sub-group, I need to show that if the group quotient $G/[G,G]$ is cyclic then $G$ is an abelian group. My attempt is to let $g\in G$ s.t $g$ ...
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Algebraic Structures that do not respect isomorphism

One of the first things a student learn in Algebra is isomorphism, and it seems many objects in algebra are defined up to isomorphism. It then comes as a mild shock (at least to me) that quotient ...
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Converse of Lagrange theorem does not hold on $S_n$ for $n\geq 4$. [closed]

I already know that converse of Lagrange theorem does not hold on $A_n$. If converse of Lagrange theorem does hold on $A_n$ , Some group of $A_n$ has all 3cycles and it is whole group. Could you ...
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Concerning Groups having the property that intersection of any two non-trivial subgroups is non-trivial

The group of rational numbers $(\mathbb Q,+)$ has an interesting property , that the intersection of any two non-trivial subgroups of this group is non-trivial . Let us call this property the " non-...
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Are the groups $\mathbb R/ \mathbb Z$ and $\mathbb R^2 / (\mathbb Z \times \{0\} )$ isomorphic?

Is it true that as groups , $\mathbb R/ \mathbb Z \cong \mathbb R^2 / (\mathbb Z \times \{0\} )$ ? I only know that $\mathbb R \cong \mathbb R^2$ (as groups ) but I can see no way to decide whether ...
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Why is the order of an element in a cyclic group a factor of the order of the cyclic group it is in?

If G has an element a of order k, then the group generated by a consists of {e, a, a^2, ..., a^(k-1)} which are all distinct elements of the group generated by a whose orders are a factor of k. I don'...
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Definition of $k$-transposition group

In A monster tale: a review on Borcherds’ proof of monstrous moonshine conjecture, a $k$-transposition group is defined as follows. Recall that a $k$-transposition group $G$ is one generated by a ...
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Studying $\operatorname{Hom}(G,\mathbb{Z}/2\mathbb{Z})$ where $G$ is finite.

Let $G$ be a finite order group. Then we can write $|G| = 2^n(2m+1)$ for some non-negative integers $n$ and $m$. I'm trying to show that $\operatorname{Hom}(G,\mathbb{Z}/2\mathbb{Z})$ is an abelian ...
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Doubly transitive and solvable

I wonder if there are subgroup of $S_n$ that are solvable and doubly transitive for $n\geq 5$. Can someone find an example or prove that they don't exist?
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Automorphism of a finite cylic $p$-group

Let $G$ be a finite group. Suppose that $A$ is a cylic $p$-subgroup and $A\unlhd N_G(Q)$ for some subgroup $Q$ of $G$. Let $x \in N_G(Q)$ be an $p'$-order element. Then $x$ induces an automorphism of ...
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Sylow-p-group of matrices group over finite field.

Let $F$ be a finite field of characteristic $p$ and $U$ a subgroup of $F^*$. Let $G$ be the Group of $n*n$ upper triangle matrices over $F$ with elements of $U$ on the diagonal. Find a Sylow-p-...
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Least symmetric group having a certain Abelian group as subgroup

Given an Abelian group $G\simeq\bigoplus_{k}\mathbb Z_{p^{n_k}_{k}}$, where $p_1\leq p_2\leq ...$ are primes, how to calculate the least symmetric group $S_n$ having a subgroup isomorphic to $G$?
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How many elements does group of symmetries of this logo have?

I could only rotate but not reflect it. So is it $2$?
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Properly discontinuous group actions - Hausdorffness

I was told to prove the following: If an action is free and satisfies that each point has a neighborhood $U$ satisfying $U \cap gU=\emptyset$ except for finitely many $g\in G$, and moreover the space ...
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When is the commutator subgroup a maximal subgroup? [closed]

Let $G$ be a group , under what conditions do we have that $G/[G,G]$ is a finite group of prime order ?
Is it true that $SL(n, \mathbb R)=<\{ABA^{-1}B^{-1} : A,B \in GL(n,\mathbb R) \}$ >? [closed]
Is it true that $\operatorname{SL}(n, \mathbb R)=\left\langle \left\{ABA^{-1}B^{-1} : A,B \in \operatorname{GL}(n,\mathbb R) \right\} \right\rangle$?