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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Prove that identity element is unique

During an exam I tried to prove that the identity element of group (G.•) is unique. I approached this way: Suppose there are two identity elements $e_1$ and $e_2$. Then: $a^{-1}•a=e_1$ ...
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1answer
65 views

Are these groups solvable?

I am thinking of Baumslag-Solitar groups of type $BS(1,m)=\langle a,b \mid bab^{-1} = a^m\rangle$ as a prototype. We can think of them as follows: Start with an infinite cyclic group $\langle ...
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0answers
19 views

The subgroup of $PGL(V)$ stabilizing a projective configuration

Let $P(V)$ be a projective space and consider the natural action of $G=PGL(V)$ on it. Let $S=\{p_1,\dots, p_k\}$ be a finite set of points in $V$ where $k\geq 2$. Is there any reference about the ...
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1answer
31 views

About the conjugacy classes of a finite group

Let $K_1, \cdots , K_n$ denote the conjugacy classes of a finite group $G$. For $x \in K_s$, define $n_{ijs} = |\{(y, z) \in K_i × K_j : yz = x\}|$. I want to show that $n_{ijs} = n_{jis}$. How ...
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132 views

Do there exist simple groups of order $n(n+1)$ for some integer $n>1$?

In the comments to this question, it was remarked that the question of whether a finite group of order $p(p+1)$ is necessarily not simple becomes fairly interesting if we do not assume that $p$ is a ...
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1answer
30 views

Automorphism Group of a cyclic p-group

I want to show that the automorphism group of $C_p^{k}$ is cyclic for an odd prime $p$. I know that the order of $Aut(C_n)$ is $\phi(n)$ and so the order of $C_{p^{k}}$ is $\phi(p^{k}) = ...
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0answers
10 views

Restriction of an isogeny is still an isogeny?

Given that $E \times F \twoheadrightarrow G$ is an isogeny where $E,F$ are both subgroups of $G$, is its restriction to a subgroup $H$ of $G$, $E \times (F \cap H) \twoheadrightarrow (G \cap H) = H$, ...
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0answers
13 views

Show that all elements of $\left<a,b,c\right>$ are of the form $a^ib^jc^k$ (comprehension)

Let $G=\left<a,b,c\right>$ a subgroup of $\mathfrak S_6$ where $a=(123),b=(456)$ and $c=(23)(45)$. Show that every element of $G$ can be uniquely written as $a^ib^jc-k$ where $0\leq i,j\leq 2$ ...
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2answers
20 views

Looking at two versions of Fundamental Abelian groups theorem

$G$ is a finite abelian group. Then it can be expressed with a direct product of cylic groups with prime power order $G$ is a finite abelian group with order $n$. Then it can be expressed as ...
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1answer
20 views

Irreducible representation of $S_3$

How can I show that this representation of $S_3$ is irreducible? $$\rho\left(e\right)=\left(\begin{array}{cc} 1 & 0\\ 0 & 1 ...
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1answer
41 views

Show that $\ker\phi = (-5+ \sqrt{-7})$

Let $\phi : \mathbb{Z}[\sqrt{-7}] \rightarrow \mathbb{Z}/32$ s.t. $\phi (a+b\sqrt{-7}) = \overline{a+5b} $. Show that $\ker\phi = (-5+ \sqrt{-7})$, where $(-5+ \sqrt{-7})$ is the ideal generated by ...
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0answers
20 views

Prove (Z8, +), (D4, * ) (the group of symmetries of the square) and the quaternion group (Q; *) are not pairwise isomorphic? [duplicate]

Can someone please help me with the following question? I've seen similar questions but am still struggling to get to grips with showing how their not pairwise isomorphic. "Prove: (Z8, +) (D4, ...
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0answers
43 views

Grigorchuk Group Virtually Nilpotent

I'm studying the Grig group and I know that it does not have polynomial growth. But I wanna proof it using the Gromov Thm. Where may I find the proof about virtually nilpotent or some tips?
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1answer
31 views

Why does Lagrange come into play here? [duplicate]

I am given this problem, and the solution reasons differently from me and obscurely, to me Let $G$ have order $30$. Show that it is not simple. The solution is apparently So, what I don't ...
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1answer
34 views

Let $G$ a group and $A,B$ two normal sub group of $G$. Prove/disprove $A\leq B\implies G/B\leq G/A$?

Let $G$ a group and $A,B$ two normal sub group of $G$ such that $A\leq B$. Do we have $$G/B\leq G/A\ \ ?$$ Or at least $G/B\subset G/A$ (as set) ? To be honest, the righting $G/B\leq G/A$ look a ...
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3answers
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Sylow theorem; Sylow $III$

Just curious, I have't seen any statement that answers this so far. And I'm not at the stage of being able to or given a proof of the theorem so I am not sure. Sylow $III$; If $G$ has order ...
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1answer
34 views

Is the centralizer of a group equal to the intersection of the centralizers of its generators?

Let $G$ be a finite group, and $H\leq G$ such that $H=\langle x,y\rangle$. Is the following true: $C_G (H) = C_G (x) \cap C_G (y)$ It seems to me the answer is yes. Given $c\in C_G (H)$ then $ch=hc$ ...
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38 views

No idea why and cannot prove

Can't find any theorem or helpful ideas that might link to this. In all honestly, I am very lost in this topic. If $H$ is finite abelian group and some $a$ such that $a|\exp(H)$ then $H$ has an ...
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What happened to the “permutation-groups” tag? [migrated]

There used to be a "permutations-groups" tag, which I don't see anymore. What happened to it? Can it be put back?
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1answer
32 views

Groups of order 12 with a normal 3-subgroup contain an element of order 6

Let $G$ be a group of order $12$ with a normal $3$-subgroup (which is unique by Sylow's theorems). Does it contain an element of order $6$? I just need a hint to prove it without classifying all the ...
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1answer
28 views

connection between conjugation and “relabeling”

If we look at $S_n$ we know that conjugation preserves cycle type, and that conjugation of some $\tau$ by $\sigma$ permutes the numbers in the cycle representation of $\tau$ through $\sigma$. The ...
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2answers
102 views

How does one prove that $\forall a \in \mathbb{Z}, \quad a^{13} = a \bmod 35$?

How does one prove that $\forall a \in \mathbb{Z}, \quad a^{13} = a \bmod 35$? I would recall that $a^p = a \bmod p$ if $p$ does not divide $a$, but $13 \neq 35$ and besides the statement should ...
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1answer
86 views

Group of order $p^2+p $ is not simple [duplicate]

Can some one please give me a hint to prove that every group of order $p^2+p$ is not simple?
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2answers
31 views

Are the normalizers of Sylow p-subgroups isomorphic?

Let $G$ be a finite group and $A,B \in \text{Syl}_p G$, for some prime $p$. Is it always true that the normalizers $N_G(A)\cong N_G(B)$? I just need a hint to get started, because I don't know where ...
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1answer
65 views

$H_1 ,H_2 \unlhd \, G$ with $H_1 \cap H_2 = \{1_G\} $. Prove every two elements in $H_1, H_2$ commute

This is the proof, which I mostly understand except for one bit: You have $h_1 \in H_1$ and $h_2 \in H_2$. We also have $h_1^{-1}(h_2^{-1}h_1h_2) \in H_1$, because $h_2^{-1}h_1h_2 \in ...
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1answer
20 views

If the intersection of two normal subgroups is trivial, then their elements commute [closed]

How to show that if $N \ \& \ M$ are 2 normal subgroups of group $G$ and $N\cap M=\{e\}$ (identity element), then for any $n\in N \ \&\ m\in M $, $nm=mn$?
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1answer
54 views

Check that two elements are conjugate

Are the matrices $\left[\begin{array}{rr} 1 & 1 \\ p & 1 \end{array}\right]$ and $\left[\begin{array}{rr} 1 & q \\ 1 & 1 \end{array}\right] $conjugate elements of $GL_2(\Bbb R)$? Are ...
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0answers
34 views

Need an very extensive explanation on what this problem is talking about

Group theory and my lecture notes says nothing about this but yet expects me to know it. I'm unfortunately not Galois or anyone around that and have no means to work what this even means on my own ...
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1answer
48 views

Is $\mathbb Z/m\mathbb Z \times\mathbb Z/n\mathbb Z$ isomophic $\mathbb Z/\operatorname{lcm}(m,n)\mathbb Z$ for every $m,n$?

I know if $(m, n)=1$ then $\mathbb Z/m\mathbb Z \times\mathbb Z/n\mathbb Z$ is isomorphic to $\mathbb Z/\operatorname{lcm}(m,n)\mathbb Z$. Is it true for all $m,n$? I want to understand the structure ...
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2answers
24 views

The point of a group-theoretic Chinese Remainder Theorem?

It states that for coprime $m,n$ nonzero integers, $C_{mn} \cong C_m \times C_n$. However, I know a theorem that says Cyclic groups with the same order are isomorphic. So $C_{mn} \cong C_m ...
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0answers
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Composition Series Analogous to Compactness?

The wikipedia page for group with operators makes the following claim about composition series as being analogous to compactness: The Jordan–Hölder theorem also holds in the context of operator ...
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0answers
18 views

Finding the generators of $SU(3)$ different from the Gell-Mann matrices?

I want to find a set of generators of SU(3) different from the Gell-Mann matrices. How should I go about it? Can I construct it in such a way that at least three of the 8 generators when squared gives ...
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0answers
28 views

Crossed homomorphism from cyclic group

Let $\langle x\rangle$ be a cyclic group, and $N$ any group. It is easy to tell when a map $x\mapsto n $ can be extended to a homomorphism: if $o(x)$ is infinite then always; if $o(x)$ is finite then ...
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1answer
33 views

Property of natural homomorphism

Let $G$ be a group and $A\unlhd G$ and $H\leq G$. Suppose that $\phi : G \rightarrow G/A$ denotes the natural homomorphism. if $HA = gHg^{-1}$A then $\phi(H) = \phi(gHg^{-1})$. I know that $\phi(HA) ...
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Equivalence of crossed homomorphisms: understanding

Let $\pi \times_{\varphi} G$ be semi-direct product with $G$ normal; $f_1,f_2\colon \pi \rightarrow G$ be crossed homomorphisms: $$f_i(\sigma\tau)=f_i(\sigma)^{\tau} f_i(\tau),\,\,\,\,\,\,\,\, \mbox{ ...
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Non-abelian groups of order $50$

Are there non-abelian groups of order $50$? If so, how many elements of order $2$ can they have? Such a group would have a unique $5$-Sylow-group, so it would have at least $25$ elements not of order ...
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2answers
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Show that the set of all elements $a$ of a group $G$ such that $ax=xa$ for every element of $x$ of $G$is a subgroup of $G$.

That is, the set of elements of a group that commute with all elements of the group is a subgroup. What is the question asking? From what I'm interpreting it as is that for all elements of $a$ that ...
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1answer
20 views

What is the notation $PQ$ supposed to mean for subgroups?

My notes just jumps to it without prior explanation like I am supposed to know it. I don't. It's talking about Sylow p-groups and such, $A$ is a group and let $P,Q$ be sylow $p,q$-groups ...
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1answer
17 views

Crossed homomorphism from semi-direct product: confusion in definition

(Ref: this) Let $\pi \times_{\varphi} G$ be semi-direct product in which $G$ is normal and $\pi$ is complement. Let $\omega$ be another complement of $G$ in above semi-direct product (so $\pi ...
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Prove that $(a,b)^2={a^2}{b^2}$ for all elements $a,b$ of a group iff the group is an abelian group.

An abelian group from what I know is that it's a commutative group. A group is a nonempty set with one associate binary operation that is closed, has a unity, and multiplicative inverses for each ...
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1answer
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Is there a rigorous way of saying, “if $G$ and $H$ are isomorphic then $G$ and $H$ share all the same properties”?

So, most of us have been in an introductory algebra course and proved basic facts about isomorphic groups (or rings, modules, etc., we'll use groups as the example and call them $G$ and $H$), such as ...
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Number of ways to connect sets of $k$ dots in a perfect $n$-gon

Let $Q(n,k)$ be the number of ways in which we can connect sets of $k$ vertices in a given perfect $n$-gon such that no two lines intersect at the interior of the $n$-gon and no vertex remains ...
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1answer
23 views

Website or book with Hasse diagrams of subgroups

I need to look at Hasse diagrams of very many groups, especially high powers of small symmetric groups. Is there any place where I could look them up? Calculating them myself would be a huge amount of ...
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Ordering of elements in the base of a group

In section 4.6.7 of HANDBOOK OF COMPUTATIONAL GROUP THEORY, the authors use an ordering $\prec$ for the elements in a coset. That ordering, $\prec$, was defined in section 4.6 as follows. Throughout ...
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1answer
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Determine the number of homomorphisms between $\Bbb{Z}_{10} \times \Bbb{Z}_{25}$ and $S_4$

EDITED ANSWER Determine the number of homomorphisms between $\Bbb{Z}_{10} \times \Bbb{Z}_{25}$ and $S_4$. Here, $\Bbb{Z}_n$ is the integers from $0$ to $n-1$ with addition modulo $n$. $S_4$ is the ...
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2answers
39 views

Show that any group of order $2p$ is soluble

Let $|G| = 2p$. The result is clear is $p$ is even. The proof goes on to show that there is only one subgroup of order $p$ is p is an odd prime - my question is that, why do we need to show that ...
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2answers
27 views

Does “order of a subgroup $n$” mean “there is an element of order $n$ in $G$”?

I am not sure about this. I understand it when it is cyclic. But if not stated so, I cannot reason as to why.($G$ here I assume finite) $G$ is a group with some subgroup $H$. Then, if $|H|=n$ then ...
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0answers
25 views

Why are the conjugated generating reflections the only reflections of a finite reflection group?

Why are the conjugated generating reflections the only reflections of a finite reflection group? Suppose $W$ is a finite reflection group. (i.e $W$ is finite and is generated by a set of orthogonal ...
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1answer
19 views

Show that a linear transformation on $\Bbb R^{2n}$ preserves the symplectic form $\Omega$ if and only if $A^T \Bbb J A = \Bbb J$

Hope everyone is well. I'm really needing some help with this question I've been doing for the matrix groups course I'm taking. Consider the skew-symmetric billinear form (on the vector space $\Bbb ...
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1answer
11 views

Subgroup with order =LCM of two subgroups

The following is a question that was asked by my teacher as a ponder-upon question:to which unfortunately I have not been able to put a single forward step. If an abelian group has subgroups of order ...