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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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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79 views

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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48 views

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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27 views

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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1answer
51 views

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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1answer
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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Extension of $A_{4}$ [closed]

Is there any finite group $G$ with $\pi(G)=\{2,3\}$, and $Z(G)=1$ such that 1) $G$ has a normal $3$-subgroup $N$; 2) $G/N$ is isomorphic to Alternating group of degree $4$.
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39 views

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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1answer
59 views

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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1answer
43 views

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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19 views

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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119 views

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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1answer
21 views

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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1answer
48 views

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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2answers
36 views

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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1answer
39 views

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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24 views

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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36 views

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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2answers
48 views

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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1answer
25 views

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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1answer
17 views

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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33 views

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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52 views

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 ...
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88 views

Let $G$ be a group and $N$ be a minimal normal subgroup of $G$, $G_p \in Syl_p(G)$, where $p$ is an odd prime divisor of $|G|$.

Let $G$ be a group and $N$ be a minimal normal subgroup of $G$, $G_p \in Syl_p(G)$, where $p$ is an odd prime divisor of $|G|$. Suppose that $M/N$ be a maximal subgroup of $G_pN/N$. then $M = PN$ for ...
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1answer
33 views

Order of product of non-disjoint cycles

Let $a$ and $b$ be two non-disjoint cycles of order $m$ and $n$. Is there any general formula for the order of $a b$? I understand that we can convert any non-disjoint cycles into disjoint cycles and ...
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1answer
32 views

On the definition of free product of groups.

Let $G$ and $H$ be groups. Their free product is a group $G*H$ with homomorphisms $\iota_G:G\rightarrow G*H$ and $\iota_H:H\rightarrow G*H$ so that given any other group $X$ with homomorphisms $f_G:G\...
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1answer
38 views

If a subgroup has smallest prime index, then it is normal

Assume that $G$ is finite with $p$ the smallest prime dividing its order. Suppose $H < G$ with $[G:H]=p$. Prove that $H \lhd G$. I've seen this question a few times on here but all the proofs I ...
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4answers
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Is it true that the order of the group is a power of $2$ if every element has order $2$?

I read in this old question that If $G$ consists only of elements of order 2, then $|G|=2^m$ for some $m$. But it's not clear to me. I tested the base case $G=\{a,b,ab,e\}$ but induction ...
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1answer
24 views

Non-cyclical behavior of a union of subgroups

Let $G_1,G_2,...$ be subgroups of a group $G$. I would like to show that if $G_i \subseteq G_{i+1},G_i \neq G_{i+1}$, then $\bigcup_{i=1}^{\infty} G_i$ is not a cyclic group. This seems like an ...
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A subgroup of $S_n$ having index 2 is $A_n$. [duplicate]

I want show that If $ n \geq 3$ , A subgroup $H$ of $S_n$ having index 2 is $A_n$. Say $H$ is such a subgroup. Since $A_n$ is a normal subgroup of $S_n$,$H$ normalizes $A_n$ . By 2nd ...
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2answers
60 views

Group whose all subgroups have infinite index

Is there a group $G$ satisfying the following conditions? If $H$ is a proper subgroup of $G$ , then $[G:H]$ has infinite index. I guess $\mathbb{Q}$ is such group.
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How is the kernel of a group action defined?

Question: Show that the kernel of the group action of $G$ acting on set $A$ is equal to the kernel of the corresponding permutation representation of this action. I'm lost in this definition as ...
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Show that $\sum_{d\mid f} \varphi(f/d) a^{|d|} \equiv 0 \pmod f$

This equation is correct when $f$ and $a$ are any integers. I want to show that this holds for $f,a\in K[x]$ where $K$ is any finite field. In the equation $\varphi(f)$ is defined as $|(K[x]/(f))^\...
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How to prove that $\langle\{ (1,2),(1,2,3) \}\rangle=\mathfrak{S}_3$

Prove that $\{ (1,2),(1,2,3) \}$ Generating set of a symmetric group $(\mathfrak{S}_3,\circ )$ SOlution provided by book we 've $(1,2,3)(1,2)(1,2,3)^2=(2,3)$ and $(1,2,3)^2(1,2)(1,2,3)=(1,3)$ ...
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Odd order subgroups of $PSL(2.q)$

Let q be an odd prime power. Is it true that every odd order subgroup of $PSL(2,q)$ is abelian ? If yes, how can it be proven ?
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1answer
23 views

If $G$ is p-nilpotent then $G$ has only one p-Sylow. Is it true?

Let be $G$ a group p-nilpotent. So $G$ has a p-normal complement $H$ that is a $p'$ Hall subgroup. I have read that if $G$ has a p-complement $H$ then this $H$ is unique. I don't understand: the p-...
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Reducibilty of a Hall system into a subgroup

Suppose that $\Sigma$ is a Hall system of $G$ and $L\leq G$. Then if $ \Sigma \cap L = \{ H\cap L | H\in \Sigma \}$ is a Hall system of $L$, we say that $\Sigma$ reduces into $L$. The following is a ...
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62 views

In a group $G$ if $x^3=e$ has more than one solution then the number of it's solutions is odd.

In a finite group $G,$ if the equation $x^3=e$ where $e$ is the identity has more than one solution, then the number of it's solutions is odd. My attempt Suppose we have even number of distinct ...
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1answer
34 views

Centralizer of a Sylow $2-$subgroup of $PSL(2,q)$

Let q be an odd prime power. By a classic result, a Sylow 2−subgroup $P$ of $SL(2,q) $ is generalized quaternion. It is an irreducible subgroup of $GL(2,q)$ (since otherwise its natural ...
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1answer
49 views

Complement of Normal subgroups and free groups

Does every normal subgroup has complement in free groups? What about free abelian groups i.e. Is free abelian gorup complemented group? Definition: If there exist a subgroup K such that HK = G and H ∩...
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$G$ nilpotent group and $N\trianglelefteq G$ then $[N,G]<N$, attempt of the proof

I need help in proving this fact: Let be $G$ nilpotent group and $N$ a normal and non trivial subgroup. Then $[N,G]$ is a proper subgroup of $N$. My attempt: I know the following fact: Let be $H$ ...
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1answer
33 views

Suppose $H$ is a subgroup of $S_n$ then does there exist $i\in \{1,2…,n\}$ such that $H=\mathbb{Stab}(i)$?

Suppose $H$ is a subgroup of order $(n-1)!$ in $S_n$ then does there exist $i\in \{1,2...,n\}$ such that $H=\mathbb{Stab}(i)$ ? My motivation behind asking this question comes from a question on ...