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.

33,541 questions
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When does a short exact sequence of groups imply it is isomorphic to direct product group

Suppose that $1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1$ is a short exact sequence of groups. Then, what is a (necessary and )sufficient condition for $G\cong N\times Q$. In other words, ...
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Vertex- and edge-transitivity, graphs

I know the theoretical definition of vertex- and edge-transitive graphs (VT/ET) . However, when given a graph I find it hard to say whether or not it is VT or ET. Can I say that a graph is VT if each ...
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Normalizer of $\langle(123), (456)\rangle\ \subset S_6$

I am trying to find the normalizer of $\langle(123), (456)\rangle\ \subset S_6$. Then only way I can think of is to check element by element whether $g \in S_6$ takes the $(123), (456)$ back to the ...
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Proof (or counter-example) of the existence of an element of order $p$ in a conjugacy class $aH$ of a subgroup $H$ of a finite group $G$

Let be $G$ a finite group, $H$ a subgroup, $p$ a prime number and $a \in G \setminus H$, such that $a^p \in H$. I know there is an element $x \in aH$ such that $x^{p^k} = e$, for some $k$ and $e$ the ...
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Calculation of trace of linear map $u\mapsto\frac{1}{|G|}\sum\limits_{g\in G} \chi(g^{-1})gu$

Let $G$ be a finite group, $V$ a $\mathbb{C}G$-module with character $\chi$. Let $z:=\frac{1}{|G|}\sum\limits_{g\in G}\chi(g^{-1})g\in\mathbb{C}G$. Let $U$ be an irreducible $\mathbb{C}G$-module with ...
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Validate my proof of $U(n) = \lbrace k : (k, n) = 1 \space and \space 0 < k < n \rbrace$ is closed under modular multiplication

Let $A, B \in U(n)$ for any $n \in \mathbb N^+$. Then we need to show that (ordinary) multiplication of $A, B$ ($AB$) satisfies the following, $$(AB, n) = 1$$ Which is can be done using the Bézout's ...
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Prove the following statement: “Let $G$ be a group, and $N ⊲ G$. Then $G/N$ forms a group under the operation $(gN)(hN) = ghN$ ” [duplicate]

I had this as a statement in my book, but I am unable to prove it using the four basic properties of a group: Closure Associativity Existence of identity Existence of inverses.
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Does every cofinite Fuchsian group contain a hyperbolic element?

Let $\Gamma\subset PSL_2(\mathbb{R})$ be a cofinite Fuchsian group (e.g. a Fuchsian group with finite fundamental domain). Does $\Gamma$ necessarily contain a hyperbolic element? At first, I tried ...
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What is the difference between: $G= \langle \{g\}\rangle$ and $G = \langle g\rangle$?

I have seen these two different notations for a cyclic group $G$ generated by an element $g\in G$. I am curious if there is a difference between the two notations mentioned above.
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Topics in Algebra - N. Herstein Exercise from Section 2.12, Question 16 (Page 103)

Please help me with this Herstein exercise (Page 103,Sec 2.12, Ques 16). \begin{array} { l } { \text { If } G \text { is a finite group and its } p \text { -Sylow subgroup } P \text { lies in the ...
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What is the symmetry group of $(z, z^*) \in\mathbb{C}^2$?

I'm curious about the symmetry group of $(z, z^*)\in\mathbb{C}^2$; that is, the complex vector in $\mathbb{C}^2$ consisting of a complex number and it's conjugate. The first thing I though of was to ...
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Show that the set of $n\times{n}$ matrices paired with matrix addition forms an infinite abelian group.

I am trying to show that the set of $n\times{n}$ matrices paired with matrix addition forms an infinite abelian group. clearly matrix addition is commutative hence is abelian. But how does one show it'...
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Show that faithful action of groups of order $128$ on sets of order $8$ are isomorphic

Let $G_1, G_2$ both be groups of order $128$, $X_1, X_2$ be sets of order $8$. $G_i$ acts on $X_i$ faithfully. Show that there exists an isomorphism $f: G_1 \cong G_2$ and a bijection $p：X_1 \cong X_2$...
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Examples of proper group actions

Lately I encountered the definition of proper group: A $G$-action on $X$ is called proper if the function $f:(g,x)\mapsto (g\cdot x, x)$ is proper, i.e. for any compact set $U\subset X\times X$, ...
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Decompose into irreducible representations the restriction of the n-dimensional irreducible representation of SU(2) [on hold]

Problem: Decompose into irreducible representations the restriction of the n-dimensional irreducible representation of SU(2) to a subgroup of diagonal matrices
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Decompose into irreducible representations a regular representation of a group S3 [on hold]

The problem is to decompose into irreducible representations a regular representation of a group S3. It has three represantations: trivial, sign, tautological. One regular representation should have ...
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let $G$ is group,that for all $n \in\mathbb{N}$, it has exactly $\phi(n)$ elements of order $n$. Then Every finite subgroup of $G$ is cyclic.

let $G$ be a group having property that for all $n \in\mathbb{N}$, it has exactly $\phi(n)$ elements of order $n$. Show that every finite subgroup of $G$ is cyclic. let $H$ is subgroup of order $m$ ...
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Naming a “split” subgroup of the symmetric group?

let $a+b = n$. I'm interested in the group that permutes first $a$ elements then the next $b$ elements. It should be a subgroup of the symmetric group $S_n$, right? For example, $a=1,b=2$, the group ...
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Efficient algorithm to determine if word vanishes on group?

Suppose we have a word in $t$ letters, and suppose we have a finite group $G$. We can view the word as a map $G^t \to G$- by plugging into our letters and then doing the multiplication in the group. ...
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Find all possible orders of the elements in a group [on hold]

Let $G$ be a group such that $|G| = 2^2 pq$, such that, $p,q$ are primes. How can I find all the possible orders of the elements in the group?
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Show isomorphism of two groups

Prove isomorphism of groups $\langle G, + , {}^{-1}\rangle$ and $\langle G, *,{}^{-1}\rangle$, where $a*b=b+a$ $\forall a,b \in G$ I'm barely starting to study abstract algebra. So how do I show ...
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Proving that a quotient is virtually a nilpotent group

Let $G$ be a group and let $L$ be a normal subgroup of $G$. Moreover, I have normal finite index subgroups $N_{i}$ of $G$ for $i\in \{1,\cdots,n\}$ such that the $n$-fold commutator [N_{1},\cdots,N_{...
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commutative law is not derivable in group theory? [on hold]

This is the last question of my homework. I solved everything else, but this gives me a hard time. Hope my translation is correct and this is the right place for this kind of question. Show that the ...
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Definig relators of group lie in its subgroup [on hold]

If $G$ is a $p-$group presented by some relations and relators; and $H$ is a subgroup of $G$ with the property that all the defining relators in the presentation of $G$ are in $H$, what can we know ...
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Same binary operator gives multiple identities on different subsets

Give me an example of a binary operation on two sets $S_1$ and $S_2$ where $S_1$ and $S_2$ are subsets of a set $S$ such that when the operation is carried on $S_1$ then it has identity $e_1$ and when ...
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I have a doubt regarding the group {4,8,12,16}under multiplication modulo 20.

I faced a question show that {4,8,12,16} is a group under multiplication mod 20.It is fine.I have solved the problem also.But I am feeling something strange in it.The idenity element of multiplication ...
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Semi group $S$, $x^2y=y=yx^2$ for all $x$, $y$ show that $S$ is abelian

I can not understand few steps please just give me hint what actually done in second part .
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Does the Burnside ring have a comultiplication?

Background: The Burnside ring $\mathrm{Burn}(G)$ can be formed by first constructing the semiring (like the naturals $\mathbb{N}$) of (isomorphism classes of) $G$-sets, where addition is disjoint ...
In Fulton & Harris - Representation Theory, in exercise 3.2 we first calculate the characters of $Sym^2(V)$ for $V$ the standard representation of $S(5)$. In 3.2(ii) it is then asked to show that ...
Suppose $G$ is a groups with order $56$ and $H$ is a normal Sylow $7$-subgroup of $H$. Let $K=Z_4\times Z_2$. Then $G\cong H\rtimes K$ for some homomorphism \$\varphi:K\rightarrow \text{Aut}(H)\cong ...