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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.

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Group algebra and polynomial algebra.

It occurred to me that since the group algebra of $\mathbb{Z}$, $k\mathbb{Z}$, has multiplication $$\left(\sum _{n\in \mathbb Z} a_n n\right)\cdotp \left(\sum _{n\in \mathbb Z} b_n n\right)=\left(\...
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1answer
18 views

Normal subgroups of invertible affine transformations $\pmod p$

Fix a prime $p$. Find the number of normal subgroups of the group $G$ of invertible affine maps $x \to bx + c$, $b \neq 0$ on $\mathbb{Z}/p\mathbb{Z}$. It is clear that the group has cardinality $p(p-...
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41 views

What are the schemes and implicit axioms in group theory?

I already know that the explicit aixoms in group theory are the properties of the binary operation on a set. But what are the schemes and implicit axioms in this theory? Thanks ahead!
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1answer
26 views

Is the center of an infinite solvable group infinite?

Is the center of every infinite solvable group infinite? I can see why this must be true for solvable groups constructed by taking semidirect products, but I don't think all infinite solvable groups ...
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1answer
35 views

The number of elements of order $p$ and $q$ in a group of order $pq$

Find the number of elements of orders $p$ and $q$ in a nonabelian group $G$ of order $pq$, with $p<q$ (both prime). Approach: Let the number of elements of order $t$ with $n(t)$. Let $n(p) = m$ ...
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1answer
36 views

normal subgroups of group of order $399$

Let $G$ be a group of order $399$. Then by Sylow, it must have a normal subgroup of order $19$, denoted by $H$. Let $N, K$ be groups of order $7$ and $3$. Then $HN$ and $HK$ are groups of order $133$ ...
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1answer
135 views

Does there exist a non-trivial group that is both perfect and complete?

A group $G$ is called perfect iff $G’ = G$. A group $G$ is called complete iff $Z(G) = \{e\}$ and $Aut(G) \cong G$. Does there exist a non-trivial group $G$, that is both perfect and complete at the ...
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27 views

group theory question regarding orders of elements

Let $G$ be a group with $y\in{G}$ and $n,r\in\mathbb{N}$. If $o(y)=n$, what is $o(y^r)$? My attempt: Let $$o(y^r)=a,$$ Then we have $$1_G=(y^r)^a=(y)^{ra}.$$ So we have that $$n\mid ra,$$ So either $...
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1answer
20 views

$\mathfrak{u}(2)=\{S \in Mat(n,\mathbb{C}) | S^{\dagger}=-S \}$ generators

I am calculating the generators of the algebra $$\mathfrak{u}(2)=\{S \in Mat(n,\mathbb{C}) | S^{\dagger}=-S \}$$ $$ S=\begin{pmatrix} a_{11} + i\,b_{11} & a_{12} + i\,b_{12} \\ a_{21} + ...
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1answer
46 views

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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1answer
21 views

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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1answer
86 views

An equivalence relation on the isomorphism classes of finite groups

Let all groups be finite. I am interested in the equivalence relation on finite groups given by $G \sim H$, if and only if, there exist $W_1,W_2,Z$ with $W_1 \cong W_2 \unlhd Z$ so that $G \oplus Z/...
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0answers
27 views

Prove set is a field? [on hold]

Prove $$\mathbb{Z}_{109}=\{0,1,2,\cdots,108\} $$ is a field. This is a question from my university tutorial exercise and is the first topic of second year. I only know first year elementary algebra ...
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1answer
29 views

Question for cyclic group [on hold]

If H is a cyclic subgroup of G, then G is cyclic. Is the proposition trivially true or false?
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1answer
82 views

Reasons behind the analogy between “order of groups” and “sets” (if it exists).

I had this question which I had to prove: Let G be a group and H,K < G. Then if H,K are finite subgroups, $$|HK| = |H||K|\div|H\cap K|$$ As I was trying to solve this question, it struck my mind ...
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1answer
52 views

Find an example of a group algebra with non-trivial solutions to $x^2=x$.

I am looking for a group-algebra with a non-trivial solution to $x^2=x$. That is to say, a solution with $x\neq 1$ and $x \neq 0$ where $1$ is the identity. We have $x\subset \mathbb{C}[G]$ for some ...
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1answer
51 views

How many irreducible representations can a group of order $12$ have?

We can decompose $12 = 12 \times 1$, $12 = 1+1+1+1+4+4$, $12=1+1+1+1+1+1+1+1+4$. Those are the only possibilies, but not enough for me to count all possibilities. One way to do this is to list all ...
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1answer
21 views

Dual group is cyclic for finite abelian cyclic group

If $G$ is a finite abelian cyclic group, and $\hat G$ be it's dual group i.e. the group of all homomorphisms of G into $\mathbb{T}^*$. Prove that the dual group is also cyclic. My attempt: If $G$ is ...
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2answers
44 views

the number of subgroups of the unit group $\mathbb{Z}_{n}^{\times}$

Is there a way to find the number of subgroups of the multiplicative group $\mathbb{Z}_{n}^{\times}$? For example, the number of subgroups of $\mathbb{Z}_{2020}^{\times}\cong\mathbb{Z}_{2}\times\...
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1answer
33 views

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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1answer
30 views

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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1answer
27 views

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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1answer
40 views

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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1answer
25 views

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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1answer
59 views

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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2answers
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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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1answer
156 views

Can a group be recovered from its order and automorphism? [duplicate]

Does there exists two non-isomorphic finite groups $G$ and $H$ of same cardinality with $Aut(G) \cong Aut(H)$ ? I know that the non-cyclic group of order $4$ i.e. $\mathbb{Z}_{2} \oplus \mathbb{Z}_{...
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Meta-procyclic groups.

Theorem. A pro-$p$ group is meta-procyclic iff it is a inverse limit of metacyclic $p$-groups. Proof. Let $G$ be a meta-procyclic pro-$p$ group with normal subgroup $N$ and let $M$ be an open normal ...
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1answer
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A question on “unlabeled Cayley graphs”

Suppose $G$ is a finite group $S \subset G$. Let's define $UCG(G, S)$ (unlabeled Cayley graph) as a finite unordered simple graph $\Gamma(V, E)$, where $V = G$ and $E = \{(x, y) \in G \times G| x \neq ...
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1answer
80 views

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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3answers
41 views

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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1answer
65 views

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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0answers
52 views

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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1answer
26 views

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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41 views

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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1answer
35 views

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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2answers
78 views

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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1answer
32 views

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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1answer
84 views

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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1answer
13 views

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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3answers
76 views

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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3answers
46 views

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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18 views

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 ...
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0answers
13 views

Fulton & Harris Exercise 3.2(ii) - Determining Number of Irreps that Sym^2(V) factors into without knowledge of the character table

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 ...
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37 views

Determining automorphisms for semidirect product

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 ...