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 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 ...
0
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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
37 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 ...
2
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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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0answers
26 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 ...
0
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1answer
39 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
49 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 ...
3
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1answer
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\...
2
votes
1answer
53 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^...
2
votes
0answers
46 views

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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2answers
91 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 ...
0
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1answer
35 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 ...
3
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4answers
76 views

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 ...
2
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1answer
26 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 ...
0
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1answer
33 views

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 ...
3
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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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2answers
49 views

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 ...
5
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0answers
64 views

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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4answers
58 views

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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0answers
21 views

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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12 views

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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1answer
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
35 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 ...
0
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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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2answers
35 views

$G$ 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 $H$ ...
0
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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 ...
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1answer
44 views

$\forall x \in Fix( \sigma ),\ \mathcal{O}(x)=\{ x\} $ and $\forall x \in supp( \sigma ), \{ x\}\subset \mathcal{O}(x)$

Let $\sigma \in \mathfrak{S}_{n}$ Show that : $$\forall x \in \operatorname{Fix}( \sigma ),\ \mathcal{O}(x)=\{ x\} \quad \rm{ and }\quad \forall x \in \operatorname{supp}( \sigma ),\ \{ x\}\...
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3answers
54 views

Why are these groups isomorphic?

I have this group of permutations: And I have this group of complex numbers: These groups are isomorphic to each other, but it seems I do not understand why. I was looking for similarities in ...
0
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1answer
31 views

Which vertex-transitive planar graphs represent non-self-intersecting polyhedra?

Consider an infinite planar graph with the following properties. Its vertices all have valence $3$. The faces all have $5$ edges. Now put it in cartesian space and require that the faces are all ...
1
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1answer
20 views

Action of $Aut(UT_3({\mathbb{F}_p}))$ on the set of non-commuting elements of $UT_3({\mathbb{F}_p})$

Assume that $G$ is the group of 3x3 unitriangular matrices over the field of $p$-elements $\mathbb{F}_p$. Furthermore, assume that the group of automorphisms of that group, $Aut(G)$, acts on $G$. Do ...
2
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1answer
40 views

Finite groups and prime divisors. Understanding how to deduce a claim from a certain proof.

In my algebra textbook, it goes like this. First, there is presented Cauchy's theorem: Let $G$ be a finite group, and let $p$ be a prime divisor of $|G|$. Then $G$ contains an element of order $p$....
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0answers
38 views

Prove that following set is a group [duplicate]

Given $\mathbb{R} \setminus \{-1\}$ with the operation $a*b=a+b+ab$, check whether is it or not a group. My solution: Associative $$(a*b)*c=(a+b+ab)*c=a+b+ab+c+(a+b+ab)c=\\ =a+b+ab+c+ac+bc+abc;$$ ...
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1answer
34 views

Subgroup of Index 2 is Normal - If $g \not \in H$, then $gH= G/H$?

I'm trying to understand the following proof: https://proofwiki.org/wiki/Subgroup_of_Index_2_is_Normal Specifically the part: If $g\not\in H$, then $gH=G/H$ as there are only two cosets and the ...
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0answers
52 views

A Sylow $2-$subgroup of SL(2,q) is irreducible

Let $q$ an odd prime power. By a classic result, a Sylow $2-$subgroup of $SL(2,q)$ is generalized quaternion. How can I show that it is an (absolutely) irreducible subgroup of $GL(2,q)$ ? I have ...
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vote
0answers
36 views

Is Cayley's theorem tight in a sense? [duplicate]

For any $n$, does there exist a group $G$ of order $n$ so that $G$ is not isomorphic to any subgroup of $S_j$ for $j<n$?
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3answers
61 views

If $G$ is cyclic, then $G=\{a^0, a^1, …\}$, but why does it have to be that $a^k=e$ for some $k$?

If $G$ is cyclic, then $G=\{a^0, a^1, ...\}$, but why does it have to be that $a^k=e$ for some $k$? I.e. that for $G$ to be cyclic, then surely the generating element would need to generate $e$ also. ...
5
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0answers
31 views

How many generators needed for Pell-equation-related group

Let $d$ be a positive integer which is not a perfect square. We have the norm multiplicative group homomorphism, $N:{\mathbb Q}[\sqrt{d}] \to {\mathbb Q}$ defined by $N(x+y\sqrt{d})=x^2-dy^2$. It ...
5
votes
2answers
121 views

Conjugates and commutators for twisty puzzles — so what?

This question isn't just rhetorical. I want to know what I'm missing. Twisty puzzle tutorials keep talking about how useful conjugates (operation sequences of the form ${XYX}^{-1}$) and commutators ($...
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0answers
61 views

Using Burnside's Lemma in GAP to handle special variations of the Rubik's Cube?

If you want to count the number of distinct positions of a standard 2x2x2 Rubik's Cube simple counting arguments will suffice: There are 8 corners, all distinct The 8 corners can be in any ...
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0answers
23 views

infinite Dihedral group is unimodular

How to show that the infinite Dihedral group is unimodular? A locally compact group G is strongly amenable if, for every open relatively compact neighborhood C of the unit which is symmetric we have ...
2
votes
0answers
58 views

All but a finite number of finite simple groups are groups of matrices over $\mathbb{F}_q$

In the introduction to this honors thesis, http://people.math.gatech.edu/~jrabinoff6/papers/building.pdf I found this statement: Matrix groups defined over the finite fields $\mathbb{F}_q$ ...
0
votes
0answers
34 views

Hall System of a finite solvable group

Let $G$ be a finite solvable group and $\pi(G)$ be the prime divisors of $G$. For each $p\in \pi(G)$, let $S$ be the Sylow $p$-complement of $G$ If $\pi\subseteq \pi(G)$, let $\pi^* = \pi(G) \...
2
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0answers
32 views

Show that there has to be an orbit with at least $3$ elements

Let $H \leq A_4$ be a subgroup of $6$ elements. $H$ acts on $X=\{1,2,3,4\}$ by $\sigma \cdot i = \sigma(i)$. Show that for $H$ to act on $X$ there has to be an orbit with at least $3$ elements. ...
2
votes
1answer
43 views

Order of subgroup generated by two cyclic subgroups in $S_6$.

Let $S_6$ be the symmetric group, and $\alpha=(13456)$ and $\beta=(132)$ be its two permutations. How can we find the order of the subgroup generated by $\alpha$ and $\beta$. SOl: $\alpha^5$=...