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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Does there exist a group G such that G is countable but collection of subgroups of G is uncountable? [duplicate]

Does there exist a group G such that G is countable but collection of subgroups of G is uncountable? I have tried with $\mathbb{Z}, \mathbb{Q}$ But I could not able to end up and later I choose ...
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1answer
43 views

Every group element is a product of elements in certain subsets

Let $G$ be a group. For $\theta \in$ Aut$(G)$ of order $2$, define $$ K:=\{ g\in G \mid \theta(g)=g \},\quad S:=\{ \theta(g)^{-1}g \mid g\in G \}.$$ My first question is: Assume there is a ...
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Show that $\phi(N) \leq H$

Let $\phi: G \to H$ be a group homomorphism and $N \leq G$ (with $G$, $N$ and $H$ groups). Show that $\phi(N) \leq H$ So this is what I did: Obviously $\phi(N)$ is a subset of H because $N$ is a ...
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2answers
41 views

what does these notation mean in abstract algebra? Z/Z_n

If you look at this wiki page under the image on top of right hand side, you see Z/8Z. What does it mean and give example if possible please thanks.
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1answer
52 views

Showing that $ab=0$ implies $ba=0$ in a ring

Question: Let $n$ be an integer greater than $1$. In a ring in which $x^{n}=x$ for all $x$, show that $ab=0$ implies $ba=0$. $(ab)^{n}={ab\cdot\cdot\cdot ab}=a(ba\cdot\cdot\cdot ba)b$ there ...
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1answer
49 views

Question about finite abelian group

Let G be an abelian group of order $mn$ where $\gcd(m,n)=1$. I proved that $mG$ and $nG$ are subgroups and that $G=mG+nG$ and now i want to prove the three things: the sum is direct, i.e. $mG\cap ...
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2answers
49 views

How to calculate permutation $(12)^{-1}(12345)(12)$ [closed]

I was wondering if someone could help me find $(12)^{-1}(12345)(12)$ I need to know this for calculating conjucacy classes and then a character table, thanks
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1answer
34 views

Why commutator subgroup is normal to G? [duplicate]

If $G$ is a group and $G'$ is generated by $\{xyx^{-1}y^{-1}|x,y\in G\}$, then $G'\trianglelefteq G$ and $G/G'$ is Abelian. At first, I thought this is easy because I thought ...
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1answer
52 views

If two matrices are path connected, so are their inverses

The set of $n\times n$ matrices can be identified with the space $\mathbb{R}^{n\times n}$. Let $G \le GL_n(\mathbb{R})$. We say that $A \in G$ and $B \in G$ are path-connected (not sure if this is ...
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444 views

What does algebraic mean?

If something is algebraic in a field, what does it mean? I don't know the correct phrasing. An element $a \in K$ is algebraic over $F$. Please can someone give some correct phrasing with a simple ...
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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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11 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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1answer
47 views

An non-decomposable FD group with infinitely many conjugacy class of a fixed order

Before, I asked about an FC (finite conjugacy group) which is not FD (with an infinite derived subgroup) and non-decomposable as a direct product $A \times B$ for two non-trivial subgroups $A$ and ...
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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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3answers
214 views

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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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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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
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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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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3answers
31 views

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

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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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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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
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
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
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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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
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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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
55 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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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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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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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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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
81 views

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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64 views

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 ...