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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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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3answers
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Why does the name “epimorphism” refer to a surjective homorphism?

The wikipedia page talks about epimorphisms with category theory in mind, but I have no experience with this and ask this question from a group theory point of view (answers from any point of view are ...
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1answer
17 views

Every $\pi$-separable group contains Hall $\pi$-subgroups

A group $G$ is $\pi$-separable if G has a subnormal series such that each factor is either a $\pi$-group or $\pi'$-group. This is a proof found in the book 'Theory of Finite Groups' by ...
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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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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 ...
16
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1answer
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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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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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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0answers
44 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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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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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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0answers
22 views

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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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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1answer
33 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
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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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
87 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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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
12 views

first homology group with coefficients in divisible group

I had (perhaps very elementary) doubt in the understanding of the computation of first homology group of a finite group over a divisible group. Let $\pi$ be a finite group of order $n$ and $D$ be a ...
2
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1answer
29 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 ...
2
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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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1answer
56 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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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
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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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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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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0answers
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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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4answers
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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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3answers
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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
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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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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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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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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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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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 ...
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1answer
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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
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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1answer
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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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Prove that $\frac{\Bbb{Z} \times \Bbb{Z}}{\langle(3,3)\rangle}$ is isomorphic to $\Bbb{Z} \times \Bbb{Z_3}$

I must prove that $\frac{\Bbb{Z} \times \Bbb{Z}}{\langle(3,3)\rangle}$ is isomorphic to $\Bbb{Z} \times \Bbb{Z_3}$. I am trying to do this using the first isomorphism theorem ie. for $\phi: G \to H$ ...
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1answer
60 views

What does being Abelian have to do at all with the proof?

I don't understand why the proof needs to consider cases that $G$ is Abelian and non-Abelian. If $|G|=p^n$ where $n>1$ then show that $G$ cannot be simple. It uses the theorem If $G$ is a ...
4
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3answers
55 views

Abelian finite group [duplicate]

This is a (maybe be simple) problem from Group Theory, but being a beginner, I am unable to take even a first step forward. Let $G$ be a finite group whose order is not divisible by $3$.Suppose that ...
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1answer
31 views

Cardinality of a power set? Or is “all subsets of a set” $\neq$ power set?

Again as usual, group theory is muddling me up. A proof of the Sylow $I$ theorem starts as follows Let $X$ be the set of all subsets of $G$ with $|A|=p^m$. where, the setting I have for Sylow ...
0
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1answer
57 views

How come $(\mathbb{R},\cdot)$ does not form a group? [duplicate]

This is probably a very obvious question but I've only just started learning group theory today. Is the reason that $(\mathbb{R},\cdot)$ does not form a group, because its operation is multiplication? ...
2
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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 ...