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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Subgroup of semidirect product

Let $G$ be a semidirect product of a normal subgroup $A$ with a subgroup $B$ and Let $H$ be a subgroup of $G$ such that $H\cap A$ is trivial. Is it true that $H$ is contained in a conjugate of $B$ ? ...
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Does there exist any infinite group $G$ whose each element is of order $2$?

Does there exist any infinite group $G$ whose each element is of order $2$? It is clear that the group should be abelian.I try to find out a suitable example to meet my purpose.But I have not yet ...
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Supersolvable groups and sylow towers

Is it true that a supesolvable group has a sylow tower? How can I construct the tower kwowing than there's a principal serier with factors of prime order? Can anyone help me with an hint or an idea? ...
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Finitely generated Ideals of finite algebras

i would really appreciate any help with this question. So, the question is: How to prove that finitely generated ideals of finite algebras over the ring F are finite over F? Thanks.
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Finding presentation of metacyclic groups.

I am reading the book "Presentation of groups" by D.L johnson and I have a doubt in page 88 - proposition 1. Why there is need to prove that $N \cong \mathbb{Z}_m$ because its given that $N=\langle x\...
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38 views

Abelian Group Structures

How can I determine all the subgroups of a commutative group, write the Hasse diagram, using Frobenius-Stickelberger Theorem and the isomorphism to $\mathbb{Z}_m$ of a cyclic group? In particular, for ...
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nonequivalence of category of Groups and category of Pointed Groups

Am I correct in thinking that the category pGrp, whose objects are pairs $(G,g)$ where $G$ is a group, $g \in G$ and $$\hom\left((G,g),(H,h) \right) = \{ \varphi: G \rightarrow H \hspace{1mm} \big\...
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Let $\vert G \vert = p^n m$ where $m < 2p$. Show that $G$ has a normal subgroup of order $p^{n-1}$ or $p^n$

Let $G$ have order $p^n m$ where p is a prime and $p \nmid m$. Suppose $m < 2p$. Show that $G$ has a normal subgroup of order $p^{n-1}$ or $p^n$. I have tried to apply the Sylow Theorems but I ...
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47 views

How to see that diagonal and tranvections matrices generate $GL_n(\mathbb{Z})$?

I'm trying to see how diagonal and transvection matrices generate $GL_n(\mathbb{Z})$. Is there any book that I can find a more detailed description of this problem? Thanks!
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Group theory - prove that $\forall x((x^{-1})^{-1}=x)$

so I got this question for homework: Prove that this property can be deduced from group theory: The inverse of an inverse is the identity: $\forall x((x^{-1})^{-1}=x)$ I tried building this ...
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finitely generated abelian group and its quotient

G is an abelian group with the following property: (*) If H is any subgroup of G then there exists a subgroup F of G such that G/H is isomorphic to F. Now I want to prove that If G is finitely ...
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Representation of diedral group $D_8$, why $\rho(a)^2=1$ if $a$ is the rotation?

I recall that $D_8=<a,b\mid a^4=b^2=1, bab=a^3>$. I have to determine all representation $\rho:D_8\longrightarrow \mathbb C^*$ of degree 1 of $D_8$. In my course it's written that since $\rho(a)^...
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Let $G$ be a finite group, $H$ be a $s$-permutable subgroup of $G$. if index $|G:H|=p$. then $H$ is normal subgroup of $G$.

$H$ is called $s$-permutable in $G$ if it permutes with every Sylow subgroup of $G$. Let $G$ be a finite group, $H$ be a $s$-permutable subgroup of $G$. if index $|G:H|=p$, where $p$ is an odd prime ...
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Every group of order 111 is cyclic. (True or False) [duplicate]

By Lagrange's theorem the possible order of the subgroups of the group will be 1 or 3 or 37 or 111 Now by Sylow's 1st theorem this group must have a subgroup of order 3 and as well as 37. Then we ...
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66 views

Is the alternating group a lie group

Is the alternating group a lie group. If so what is the lie algebra corresponding to it? This is not a homework questions. I need the dimension of the lie algebra (if one exists) to prove some ...
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Example of an abelian group $G$ with $A \le G$ but no $B \le G$ with $G = A \oplus B$.

I just read that if $G$ is an abelian group with subgroup $A$, then we could not always find a subgroup $B$ such that $G = A \oplus B$. I tried to come up with an example, let $G = \mathbb Z^{\mathbb ...
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Help understanding that the stabilizer is the normalizer of this subset of this group

Let $S = \mathcal{P}(G)$ $S$ be the collection of subsets in the group $G$. Let $G$ act on $S$ by conjugation. So, $g: B\to gBg^{-1}$ where $B \subset G$. This text claims that it is easy to check ...
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System normalizer of a group

If $\Sigma$ is a Hall system of a group $G$, the subgroup $N_G(\Sigma) = \{g\in G | H = H^g \;\text{for all} \;H\in \Sigma \}$ is called the normalizer of $\Sigma$ Suppose that $G$ is a finite ...
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Suppose $G$ is a finite group with nontrivial center $C$, then does $G/C$ have trivial center? [closed]

Suppose $G$ is a finite group with nontrivial center $C$, then does $G/C$ have trivial center ?
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A question on identifying normal subgroups of $S_4$

Let $H=\{e,(1 2)(3 4)\}$ and $K=\{e,(12)(34),(13)(24),(14)(23)\}$ be subgroups of $S_4$, where $e$ denotes the identity element of $S_4$. Which of the following are correct? $H$ and $K$ are normal ...
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30 views

subgroups of order 10 of a Group of order 30

I have this group of order 30 that has a normal subgroup of order 7 such that $G/N$ is isomorphic to $S_{3}$. How can I find the number of subgroups of order 10 and figure out which one is normal.
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Non-abelian group with infinitely many abelian subgroups

I'm looking for a non-abelian group which has infinitely many abelian subgroups. Do you know any examples of such groups?
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Gap: Constructing group from power commutator presentation without supplying trivial relations

Suppose I have a p-group and I want to investigate it in GAP and that I have a power commutator presentation written down for the group on paper, then what is the best way for me to construct this ...
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External Semidirect product and isomorphism

Let G and K be two groups and $\phi_1$ and $\phi_2: G \rightarrow Aut(K)$ be homomorphism. Q1: If $\phi_1$ not trivial homomorphism, can When can semidirect product of G and K using $\phi_1$ ...
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What are the conjugacy classes of the group $\{k,r \,| \, k^3=r^2=(kr)^4=1\}$?

How can I determine the conjugacy classes of a group if I have the presentation of the group? For example, we know that $S_4$ has the presentation $$ S_4=\{k,r \,| \, k^3=r^2=(kr)^4=1\}. $$ What are ...
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Index of intersection of two normal subgroups

Question: Let $G$ be a finite group and $H$ and $K$ are normal subgroup of $G$. If $[G:H]=2$ and $[G:K]=3$, determine $[G:H∩K]$. My attempt: I think $[G:H∩K] = 6$. Since $G$, $H$, $K$ are ...
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minimal subgroup is a direct product of simple groups

On page 112 of Dixon and Mortimer's Permutation Groups, Theoerm 4.3A (iii) says that every minimal normal subgroup $K$ of $G$ is a direct product $K=T_1 \times \cdots \times T_k$ where $T_i$ are ...
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Dihedral groups: Relationship between symmetries and rigid motions

In Dummit & Foote, $D_{2n},\ n \geqslant 3$ is the set of symmetries of a regular $n$-gon, where a symmetry is a rigid motion of the $n$-gon which can be effected by taking a copy of the $n$-gon, ...
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Properties of Characteristic Subgroups

I am new to characteristic subgroups, and have a theorem without a proof I don't understand yet. It says that if $A$ is characteristic in a group $G$ and $B$ is characteristic in $G$, then both $AB$ ...
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Is there a non trivial normal subgroup of a group $G$, where $|G|=pm, \ \gcd(p,m)=1$?

In fact, the exercise I'm in is this: Suppose you have an irreducible polynomial $f(x)\in \mathbb{Q}[x]$ of degree $p$, where $p$ is a prime. Also suppose that $K$ is a splitting field of $f(x)$ over ...
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A question in the proof of Counting Theorem

I am trying to follow a proof but keep getting stuck. Here is the statement of the theorem... Let $G$ be a finite group which acts on a set $X$. Let $X^g$ represent the subset of $X$ consisting of ...
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Finding a 2-cocycle in $H^2(S_3, C_4)$

As far as I know, it holds $H^2(S_3, C_4)\cong C_2$ for the trivial operation of $C_4$ as a $S_3$-module. I have tried getting a $2$-cocycle (which is not a $2$-coboundary) by its defining equation: ...
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25 views

Group of all upper triangular matrices and lower triangular invertible matrices are conjugates?

Let $X$ be set of all upper triangular matrices in $GL_n(\mathbb{R})$. Then does there exist $T\in GL_n(\mathbb{R})$ such that $TxT^{-1}$ is a lower triangular matrix $\forall x\in X$ ?
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Decompose and compute the sign of $\sigma(k)=n+1-k$

Let $n\geq 2$ and $\sigma$ is permutationof $\{1,2,\ldots,n \}$ defined by : $$\sigma(k)=n+1-k$$ Decompose permutation $\sigma$ into product of disjoint transpositions and compute the sign of it ?...
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How to prove that $\langle\{ (1,2),(1,2,\ldots,n) \}\rangle=\mathfrak{S}_n$

Let $n\geq 2, \tau=(1,2),\ c=(1,2,\ldots,n)$ two permutation of $\mathfrak{S}_n$ Prove that $$\biggl\langle\{ (1,2),(1,2,\ldots,n) \}\biggr\rangle=\mathfrak{S}_n$$ Indeed, normally i will ...
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Does there exist $A$ of infinite order in $\{ A \in GL_2(\mathbb{R}) : A^T = A^{-1} \}$?

Does there or does there not exist $A$ of infinite order in $\{ A \in GL_2(\mathbb{R}) : A^T = A^{-1} \}$? I know that elements of the form $$A=\begin{bmatrix}\cos(\theta) & \sin(\theta)\\\pm\sin(\...
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Tensor invariants constructed from identity tensor

It is evident that tensors constructed from copies of the identity tensor (and scalars) eg $t^{ij}_{kl} = 2 \delta^i_k \delta^j_l - \delta^i_l \delta^j_k$ are invariant under any matrix group, and ...
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Is the group $G_{n,\phi} = \langle x_1 , \dots, x_n \mid x_i^2, (x_i x_j)^4, x_i x_{\phi(i)} x_{i+1} x_{\phi(i)} \rangle$ abelian?

I am working on a family of finitely presented groups and I asking me the following question. Let $\phi$ be an application from $\left\{1,\dots,n\right\}$ to $\left\{1,\dots,n\right\}$ (not necessary ...
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Normalizer of $H_0 \times H_m$ in $S_{2n}$

Let $H_0 \subsetneq \dotso \subsetneq H_m$ be a chain of subgroups of $S_n$ (the symmetric group of $n$ elements) such that holds: $N(H_i) = H_{i+1}$ and $N(H_m) = H_m$. I want to prove that the ...
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Number of elements not equal to their inverses is even number

In any finite group, number of elements not equal to their own inverses is even number In my book they have paired elements with their inverses, being elements and inverses different from each other. ...
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Group theory question proving associativity

I am doing this group theory question: I have already proven that * is commutative, however, I'm I bit confused about proving for associativity. I used three variables a, b and c and said: RTF ...
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Necessary and sufficient condition for a normal group to be kernel of a homomorphism from the group to itself

I am looking for a necessary and sufficient condition for a subgroup $K$ of a group $G$ to be kernel of a homomorphism $\phi$ from $G$ to $G$. The tools that come into my mind is first isomorphism ...
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By which the Heisenberg group is introduced? [closed]

I want to know who was the first who introduced the Heisenberg group and in what year. In the Wikipedia there is just an indication that this group was named in honor of the famous German physicist ...
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What does congruency mean in $D_4$?

What does congruency mean in $D_4$? How can I check for example that For $K = \{k_0, k_2\}$, $$p_x \equiv p_y \pmod K$$ I.e. how to evaluate $(p_x - p_y) \bmod K$, specifically what is $(p_x - p_y)...
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Prove that a normal subgroup $K$ is characteristic in a finite group $G$ if $\gcd(|K|, |G/K|) = 1$

This is a part of an exercise $2.2$ on a page $203$ in a book "Algebra: Chapter 0" by P.Aluffi. Let $K$ be a normal subgroup of a finite group $G$, and assume that $|K|$ and $|G/K|$ are relatively ...
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Counting the number of distinct elements in Sylow subgroups if $|G|=30$

I'm trying to prove that if $|G|=30=2\cdot 3\cdot 5$ then $G$ has a normal $3$-Sylow or a normal $5$-Sylow. By the Sylow theorems, we would argue by contradiction in order to prove it cannot be $n_3=...
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Restricted linear representations of abelian groups

If $G$ is a group (say finite for simplicity though the question applies to infinite groups as well), what can one say about the subgroup $G^*_n = \text{Hom}(G, \mu_n)$ of the group of all linear ...
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Which groups have only real representations?

An irreducible representation $\rho$ (with character $\chi$) of a finite group is called a "real" representation if its Frobenius-Schur indicator is 1: $$\frac{1}{\lvert G \rvert} \sum_{g \in G} \chi\...
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Conjugacy classes of $\mathcal D_{10}$.

I was wondering if there is a special technique to find the conjugacy classes of $\mathcal D_{10}=\left<a,b\mid a^5=b^2=1,bab^{-1}=a^{-1}\right>$, and of $\mathcal D_{2n}=\left<a,b\mid a^n=b^...
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Proving Schur's lemma

Schur's Lemma: Let $(\Pi_i,V_i)$, $i=1, 2$ be two irreducible representations of a group $G$, and let $\phi : V_1 \to V_2$ be an intertwiner. Then either $\phi = 0$ or $\phi$ is a vector space ...