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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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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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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60 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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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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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
53 views

The relation between quasi-permutation matrix and permutation matrix? [closed]

We know that a quasi-permutation matrix is a square matrix over the complex numbers with non-negative integral trace. Can anyone tell me why it is called "quasi-permutation matrix"? Is there any ...
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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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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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38 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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30 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 ...
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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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117 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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2answers
44 views

Group of order $54$ has normal sugroup of order $27.$

Let $G$ be a group of order $54$. Prove that there exists a normal subgroup of order $27.$ Is this normal subgroup unique? Thoughts. Since $27$ divides $54$, by Lagrange's theorem we can not exclude ...
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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^...
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89 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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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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Is every quotient of a finite abelian group $G$ isomorphic to some subgroup of $G$?

I'm having difficulty with exercise 1.43 of Lang's Algebra. The question states Let $H$ be a subgroup of a finite abelian group $G$. Show that $G$ has a subgroup that is isomorphic to $G/H$. ...
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Intersection of 2 subgroups must be a subgroup proof.

Ok we have 2 subgroups of G defined as $H_1$ and $H_2$ the question wants us to prove $H_1$ intersect $H_2$ must also be a subgroup of G. This seems fairly intuitive, making as math usually hard to ...
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1answer
107 views

Primitive Wreath Product action

I am a little confused about the primitive action of the wreath product as when to use the inverse and whether to use left or right action. Let $H, K$ be groups and $K$ acts on $\Delta$, the wreath ...
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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
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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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
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 ...
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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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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
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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1answer
31 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 ...
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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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2answers
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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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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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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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116 views

Generators of the symmetric group

I know that for $n\geq2$, $S_n$ (the symmetric group of $n$ symbols) can be generated by only two elements, among which one is a $n$-cycle and other one is a transposition. But is it true that ...
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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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A4 has no subgroup of order 6

Can a kind algebraist offer an improvement to this sketch of a proof? Show that $A_4$ has no subgroup of order 6. Note, $|A_4|= 4!/2 =12$. Suppose $A_4>H, |H|=6$. Then $|A_4/H| = [A_4:H]=2$....
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Proof that the following multiplicative groups modulo m are cyclic

In the Wikipedia page about the multiplicative groups modulo $m$, the following claim is made: The group $(\mathbb{Z}/m\mathbb{Z})^*$ is cyclic if and only if $m=1, 2, 4, p^k$ or $2p^k$, where $p$ is ...
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$\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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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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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 $H$ ...
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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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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 ...
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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 ...