# 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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### Is there a general strategy for identifying the automorphism group of a graph?

I understand what an automorphism is, and I can sort of wrap my head around the idea that the set of automorphisms under composition form a group, but when asked to actually find the automorphism ...
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### Listing elements of subgroup generated by $\{12,42\}$ in the integers with addition

The subgroup generated by these elements should contain both $12\mathbb{Z}$ and $42\mathbb{Z}$ but also ideals of the form $$(12k+42j)\mathbb{Z},\;j,k\in\mathbb{Z}$$ Is this the best answer I can ...
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### are there $E(x,G)$ and $E(H,G)$ two subgroups of $G$? [closed]

let $G$ be a group and $H$ be a subgroup of $G$. let $E(x,G) = \{g \in G ; [g,x,x]=1 \}$ and $E(H,G) = \{ g \in G ; [g,x,x] = 1 ~~~ \forall x \in H \}$ . are there $E(x,G)$ and $E(H,G)$ two ...
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### Amazing isomorphisms [closed]

Just as a recreational topic, what group/ring/other algebraic structure isomorphisms you know that seem unusual, or downright unintuitive? Are there such structures which we don't yet know whether ...
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### Centralizers of Elements in the Free Group

Let $F_n$ be the nonabelian free group on $n$ generators. According to what I have been reading from various sources online, the centralizer of some element $h \in F_n$, denoted as $C_{F_n}(h)$, is an ...
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### To prove: The intersection of all normal subgroups of finite index of a free group is trivial.

Let $S$ be a set, $G$ its free group and $\mathcal{N}$ the set of normal $N \leq G$ such that $[G:N] \leq \infty$. Prove that $$\bigcap_{N \in \mathcal{N}} N \ = \ \{ e\}$$ I know how free ...
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### Why is SU(2) not the same group as T3?

Why is $S^3 = SU(2)$ not the same group as $S^1 \times S^1 \times S^1$? It seems that, the $T^3$ torus is abelian, wheras $S^3$ is not. Is that enough? Problem 2.6.5 from John Stillwell, Naive ...
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### Showing $NH$ is a subgroup of $G$

Question : If $N$ is a normal subgroup of $G$ and $H$ is a subgroup of $G$, prove that $NH$ is a subgroup of $G$. Thread is constructed on a mobile so I will attempt to be as succinct as possible. ...
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### A homomorphism $f:S_{n} \rightarrow \mathbb{R}$, prove that $f$ must be trivial.

A homomorphism $f:S_{n} \rightarrow \mathbb{R}$, prove that $f$ must be trivial, i.e., $\ f(\sigma) = 0 \ \forall \sigma \in S_{n}$. I started off by thinking that it was something to do with the ...
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### Group of order $pq$ with $p<q$ prime has $q-1$ elements of order $q$

If $p<q$ are primes and $G$ is a group of order $pq$, then $G$ has exactly $q-1$ elements of order $q$. My attempt was: Use Sylow theorem to show that $G$ has only one subgroup of order $q$, so ...
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### Counting maximal subgroups in a finite $p$-group

Let $G$ be a finite $p$-group. I want to show that if the number of maximal subgroups is strictly less than $p+1$ then $G$ is cyclic. This may not be true, but if the number of maximal subgroups is ...
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### On the relationship between $\text{SL}_2(5)$ and $A_5$ [duplicate]

I have two questions. What is the quickest way to see from scratch that $\text{SL}_2(5)/\{\pm I\}$ is isomorphic to the alternating group $A_5$? Does $\text{SL}_2(5)$ have any subgroups isomorphic ...
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### Finite subgroup of $\text{SO}(3)$ acts on set of points on unit sphere in $\mathbb{R}^3$ which are fixed via some nontrivial rotation in $G$

Let $G$ be a finite nontrivial subgroup of $\text{SO}(3)$. Let $X$ be the set of points on the unit sphere in $\mathbb{R}^3$ which are fixed by some nontrivial rotation in $G$. I have two questions. ...
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### Is there a $G$-equivariant bijection $h: G/X \to G/Y$?

Let $X$ and $Y$ be subgroups of a group $G$ such that there are $G$-equivariant maps $f: G/X \to G/Y$ and $g: G/Y \to G/X$. Is there a $G$-equivariant bijection $h: G/X \to G/Y$?
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### Derived graph of antipodal graph

Smith proposed a way of transforming an antipodal graph into a non-antipodal graph in the following way: Suppose $\Gamma$ is an antipodal graph, then it is imprimitive by a theorem of Smith. Then ...
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### Finitely generated module: terminology.

What's the meaning of the expression: $S$ is a subring of $\mathbb{C}$ finitely generated as $\mathbb{Z}$-module? Maybe that the additive group of the ring $S$ is a finitely generated abelian group? ...
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### Question on the quotient of a normaliser and a commutant

Let G be a group and $H\subset G$. Let $N_G(H)$ and $\text{Comm}_G(H)$ be the normaliser and commutant of H in G respectively. If the quotient group $K \equiv N_G(H)/\text{Comm}_G(H)$ is known and so ...
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### How many distinct ways are there to $2$-color the $8$ vertices of a cube?

How many distinct ways are there to $2$-color the $8$ vertices of a cube, with colorings only considered distinct up to rotation?
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### Trying to show that any group of order four is either cyclic or isomorphic to V

I know the question has already been asked. But I have trouble with the answer. Having a non-cyclic group $\,G=\{1,a,b,c\}\,$, how can I show that $ab=c$? In my attempt, I assume that $ab=1$, and ...
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### Problem in understanding some steps in proof

The problem here is about categorical construction of free groups, as in Lang's algebra (p.66-68). Theorem: For any set $S$, there exists free group $(F,f)$ determined by $S$ (here $f:S\rightarrow F$)...
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### Monomorphisms, epimorphisms and isomorphisms of groups category

I would like to solve the Problem 2.12 in Szekeres, A Course in Modern Mathematical Physics: Show that the class of groups as objects with homomorphisms between groups as morphisms forms a ...
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I have a group $M$, which is a large matrix group. Using the command $\texttt{LMGSolubleRadical(M)}$ I obtained $R\cong M$, where $R$ is of type $\texttt{Grppc}$, and an isomorphism $M\to R$ called $\... 1answer 22 views ### Distinct coset representative and stabilizing an element. Let$G < S_n$be a permutation group of degree$n$, and let$G^{(i)},0 < i\leq n$, be the pointwise stabilizer of$\{1,2.., i\}$in$G$. We set$G^{(0)}= G$. For$0 < i\leq n$, let$U_i$be ... 0answers 28 views ### Number of elements of Complete Right Transveral in pointwise stabilizer. [closed] Let$G < S_n$be a permutation group of degree$n$, and let$G^{(i)},0 < i\leq n$, be the pointwise stabilizer of$\{1,2.., i\}$in$G$. We set$G^{(0)}= G$. For$0 < i\leq n$, let$U_i$be ... 1answer 27 views ### Schur Multipliers in Finite Simple Groups I heard that Schur multiplier's played important role in classification of finite simple groups. By means of simple example, can one illustrate how the Schur multiplies played their role in the ... 1answer 20 views ### Relationship between elements in an internal direct product Question: Let$H$and$K$be subgroups of a group$G$. If$G=HK$and$g=h\bar {k} $, where$h\in G $and$\bar {k} $. Is there any relationship among$|g|$,$|h|$and$|\bar {k}|$? What if$...
There is an excercise in Rotman's Introduction to the theory of groups: which of the following properties, when enjoyed by both $K$ and $Q$, is also enjoyed by every extension of $K$ by $Q$? finite \$...