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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145 views

Infinite groups such that $G/G'$ has odd order.

Can someone give examples of an infinite group $G$ such that $G/G'$ has odd order.
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3answers
309 views

How to show that no. of elements $x$ of group $G$ such that $x^3=e$ is odd?

Let G be a finite group G. Then How can I show that no. of elements $x$ of group $G$ such that $x^3=e$ is odd ? I read this question in an Algebra book. Since $e^3=e$, e must be one of those elements. ...
3
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1answer
181 views

How to prove “a group $G$ of order $72$ can't be a simple group”?

By using Sylow theorem, I can prove that $G$ has either $1$ Sylow $3$-subgroup or $4$ Sylow $3$-subgroup, but I don't know how to continue the proof.
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1answer
139 views

What can we say about groups $G$ with $H_3(G)=0$?

Let $G$ be a group. What can we say about groups such that $H_3(G)=0$? If a characterization is not possible, then knowing examples of such groups would be good? Any help is appreciated. Thanks
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1answer
121 views

examples of polyclic groups

From the notes Coarse differentiation and the geometry of polycyclic groups, I found a theorem $\Gamma$ is polycyclic iff $\Gamma$ is a lattice in a solvable unimodular lie group $G$ - Mostow ...
9
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1answer
359 views

Finite Subgroups of $GL_2(\mathbb Q)$

I want to prove that the only finite subgroups of $GL_2(\mathbb Q)$ are $C_1, C_2, C_3, C_4, C_6, V_4, D_6, D_8,$ and $D_{12}$. First, we determine all possible finite orders of elements. Now, an ...
5
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3answers
60 views

Show that: $\frac{D_n}{\langle a\rangle}\simeq\mathbb{Z_2}$

Show that: $$\frac{D_n}{\langle a\rangle}\simeq\mathbb{Z_2}$$ where $D_n$ is dihedral group and $a$ is generator of order $n$.
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2answers
235 views

Having trouble understanding what ker f is

I am trying to understand what exactly 'ker f' is. My guess is that it is a set of all of the elements that were lost in the process of a mapping. For example: $f:A\to B$ $f = \left\{ \begin{align} ...
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3answers
98 views

Monotone Union of subgroups being subgroup

I saw this exercise in a book: Suppose $G$ is a group. Suppose $T$ is an index set and for each $t \in T$, $H_t$ is a subgroup of $G$. Furthermore, if $\{H_t\}$ is a monotonic collection, show that ...
3
votes
1answer
86 views

It is true that every group that has a finite number of subgroups is finite? [duplicate]

It is true that every group that has a finite number of subgroups is finite? I think not, but I can not find counterexamples.
23
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1answer
529 views

Showing that $ϕ(x)=x^n$ is a homomorphism from $G\to Z(G)$

Let $G$ be a group with $|G:Z(G)|=n$ then $\phi(x)=x^n$ is a homomorphism from $G$ to $Z(G)$. I guess it has a proof using transfer theory, I wonder whether it has an elemantary proof or not. ...
6
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2answers
181 views

Are all Sylow 2-subgroups in $S_4$ isomorphic to $D_4$?

I was assigned to show that every Sylow 2-subgroups in $S_4$ is isomorphic to $D_4$. So I figured, since $|S_4|=24=2^3\cdot 3$, every Sylow 2-subgroup has either the form: $\langle ...
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1answer
35 views

Understanding the elements in groups(modulo, cyclic and other(?))

Question exactly as given on past exam: Consider the group $G=(\mathbb{Z}/15\mathbb{Z})^\times$ (under multiplication). Let $H$ be the subgroup generated by $2$ (that is, $H = \langle 2 \rangle$). ...
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2answers
83 views

A question on Groups and its center

If $G$ be a group of order $8$ and $o(x)=4$ then how to prove that $x^2 \in Z(G)$ ? I can only figure out that $x^2=x^{-2}$ ; Please help
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0answers
48 views

Monotonic Union of Submodule

The theorem is: Suppose $M$ is an $R$-module, $T$ is an index set, and for each $t \in T$, $N_t$ is a submodule of $M$. If $\{N_t\}$ is a monotonic collection, $\cup_{t\in T} N_t$ is a submodule. Is ...
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1answer
43 views

Order of some $\alpha$ in $S_4$

I want to work out the order of $\alpha = \left[ \begin{align} & 1&2&&3&&4&\\&4&2&&1&&3& \end{align} \right]$ in $S_4$ Now when I think of ...
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1answer
68 views

on automorphisms groups a finite 2-group

Let $G=\langle a,b\mid a^4=b^4=1, bab^{-1}=a^3\rangle$. Please prove that $Aut(G)$ is generated by the automorphisms $$a\mapsto ab,\hspace{10pt} a\mapsto a^3,\hspace{10pt} a\mapsto ab^2, ...
3
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1answer
85 views

Bounded almost-homomorphisms on the integers

Let $f : \mathbb{Z} \to \mathbb{Z}$ be an "almost-homomorphism": The set $\{f(n+m)-f(n)-f(m) : n,m \in \mathbb{Z}\}$ is bounded. We may assume that $f$ is odd, i.e. $f(-n)=-f(n)$ for all $n$. Assume ...
2
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1answer
46 views

Isomorphism between groups of $2 \times 2$ matrices

I'm stuck on this problem: For $\mu \in \mathbb{R} \setminus \{1\}$ let $$G_\mu := \left\{\begin{pmatrix}a & b \\ 0 & a^\mu \end{pmatrix} : a \in \mathbb{R}^+, \; b \in \mathbb{R}\right\} .$$ ...
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5answers
238 views

Mapping from $\{1,\ldots,n!\}$ to the symmetric group $S_n$

Is there an easy known bijective mapping formula between the set $\{1,\ldots,n!\}$ and the symmetric group $S_n$? I want to pick a number $k \in \{1,\ldots ,n!\}$ and assign a unique permutation of ...
5
votes
1answer
96 views

Recovering a group action from sizes of orbits of individual elements

Let $G$ be a group (say, finite) and let it act on a set $X$ (say, also finite). For every element $g \in G$, we can consider its action on $X$. My rather vague question is What information about ...
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2answers
1k views

Finding the centralizer of a permutation

I need to find the centralizer of the permutation $\sigma=(1 2 3 ... n)\in S_n$. I know that: $C_{S_n}(\sigma)=\left\{\tau \in S_n|\text{ } \tau\sigma\tau^{-1}=\sigma\right\}$ In other words, that the ...
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2answers
79 views

Conjugates of the upper triangular matices

It's a shame...I want to give an explicit description of the set $\bigcap_{m \in GL_n(K)} mUm^{-1}$, $U$ being the upper triangular subgroup of $GL_n(K)$. It seems to be $K^\times I_n$ but I do not ...
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2answers
52 views

Almost Disjoint Uncountable Subgroups of $\mathbb{R}$

This idea struck me as I was heading to bed last night: Can we find uncountable subgroups $H,G < \mathbb{R}$ such that $H \cap G = \{0\}$? The issue I see is that uncountable sets are really big, ...
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1answer
83 views

Normal closure in a perfect group

Let $G$ be a perfect group and $H$ the conjugate closure of $B\le G$. Is there any way to tell when $H$ is perfect too? Thanks!
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4answers
355 views

Show that * is associative

Could you show me how to prove the following to be associative? Please take me through the process step by step. $$a*b=a+b+2ab$$ Where $*$ is a binary operation and $a$ and $b$ are real numbers. I ...
4
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1answer
244 views

Understanding the proof of $|ST||S\cap T| = |S||T|$ where $S, T$ are subgroups of a finite group

I'm trying to understand the proof of the following theorem: Theorem 2.20 (Product Formula). If $S$ and $T$ are subgroups of a finite group $G$, then $$|ST|\, |S \cap T| = |S|\,|T|.$$ Remark. ...
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1answer
82 views

Example of non-Abelianness of symmetric group for graphs

I know that for $n \ge 3$, $S_n$ is non-Abelian. I would like to work out an example in terms of graphs so to make it sure that I understand it right. A symmetric group of graphs of four vertices, ...
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2answers
418 views

Applications of groups, rings and modules in life [closed]

I have read calculus. Now I'm reading algebra. What is the application of groups, rings and modules in life? When we study derivations we know several applications. What about groups, rings and ...
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0answers
40 views

Quotient by a finite group of fixed-point-free isometries

I'm reading one paper in Riemannian manifolds which very briefly mentions that quotients of $S^{2}\times R^{1}$ by a finite group of fixed point free isometries include $S^{2}\times S^{1}$, ...
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1answer
89 views

FC groups with infinite derived subgroup which are not constructed by direct product of finite groups.

A group $G$ is called FC (finite conjugacny) if every conjugacy class $C$ of $G$ has a finite order. It is called FD if the derived subgroup (constructed by commutators) is finite. It is clear that ...
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3answers
63 views

Which of the following groups are isomorphic to each other?

Which of the following groups are isomorphic? $$1:(\mathbb Q_{\ge 0} ,\times , 1), \quad 2:(\mathbb R ,+ , 0),\quad 3:(\mathbb R \setminus \{0\} ,\times , 1) , \quad 4:(\mathbb C \setminus ...
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4answers
331 views

How to show a group is cyclic?

One question asking if $\mathbb{Z}^*_{21}$ is cyclic. I know that the cyclic group must have a generator which can generate all of the elements within the group. But does this kind of question ...
3
votes
3answers
45 views

Derived Subgroup and Factor Groups

Let $N \unlhd G$, does it holds that $(G/N)' = G'N /N$, or more generally for any subgroup $H \le G$ we have $(HN/N)' = H'N/N$? Does anyone has a proof of this fact? PS: Side-Question: Is it wrong to ...
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1answer
35 views

The group morphism of tye ring

Let $(G,+,\cdot)$ $(H,+,\cdot)$ be rings, we suppose that the unites $(G*,\cdot)$ and $(H*,\cdot)$ form groups respectively, for example, the matrix ring $M(n,\mathbb{R})$. There is a group morphism ...
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2answers
57 views

What can I say about the quotient group?

Let $G$ be a group of order $24$, and let $H$ be a normal subgroup of order $6$. So the quotient group $ {G\over H} $ is Abelian group?. What can I say about the quotient group beside her order?
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1answer
52 views

A question on the study of quotient groups

In dummit and foote, it was stated that "The study of homomorphic images of G is thus equivalent to study the quotient group. I am not sure whether I understood this statement correctly. I knew if I ...
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0answers
36 views

Poincare polynomial of a finite $G$-module with $G$ being a $p$-group

Recently, I've been reading Shatz's book, profinite groups, arithmetic, and geometry. Let $G$ be a finite $p$-group and $A$ a finite $G$-module such that $pA=(0)$. In the proof of Theorem 19 (p.82), ...
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0answers
274 views

Pushout and pullback of short exact sequence of groups

I think that there might be some textbooks which introduce the notions of pushout and pullback of a short exact sequence of groups. However, I cannot find any of them. To be precise, for a given ...
3
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1answer
487 views

Subgroup structure of dihedral groups

I am currently looking into structure of dihedral groups; I am interested in their subgroup structure. Dihedral group have two kinds of elements; I will use their geometric meaning and call them ...
2
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1answer
107 views

Decomposing direct product of irreps

I know characters of two 2-dimentional irreps (U and V) of a group with 6 conjugate classes. The characters are: $\begin{pmatrix} 2&-1&-1&2&0&0\end{pmatrix}$ and $\begin{pmatrix} ...
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1answer
36 views

Is $H/H_0 = HG/H_0G$?

If I have a quotient group $H/H_0$ and another group $G$ such that $H_0G\unlhd HG$. Is it then true that $H/H_0 = HG/H_0G$?
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2answers
91 views

What is the remainder after dividing $(177 + 10^{15})^{166}$ by $1003 = 17 \cdot 59$

What is the remainder after dividing $(177 + 10^{15})^{166}$ by $1003 = 17 \cdot 59$? I have made several observations about the problem but I cant see how they lead me to an answer. Observation one: ...
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3answers
224 views

Two 3-cycles generate $A_5$

I want to solve the following exercise, from Dummit & Foote's Abstract Algebra Let $x$ and $y$ be distinct 3-cycles in $S_5$ with $x \neq y^{-1} $. Prove that if $x$ and $y$ do not fix a ...
2
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1answer
48 views

The largest normal subgroup of odd order in the centraliser

For a group $G$ denote by $O_{2'}(G)$ the largest normal subgroup of $G$ of odd order. Now let $N = O^{2'}(G)$ be the smallest normal subgroup of $G$ such that $G/N$ has odd order, and $H \le G$ be ...
3
votes
4answers
111 views

An elementary problem related to $Z(G)$

Let $G$ be a group, If $xy\in Z(G)$ then $C_G(x)=C_G(y)$. Note: Even it is very elementary, I liked it. Edit: Thanks for different solutions, you may want to examine the case if $xyz\in Z(G)$ then ...
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1answer
32 views

Normal closure of the nonnormal factor of Holomorph of a Cyclic group

Let $C_n$ be the cyclic group of order $n$. Then, we can consider the holomorph $G=C_n\rtimes Aut(C_n)$. let $H$ be such that $Aut(C_n)\leq H\trianglelefteq G$. Is it necessarily the case that $H$ is ...
1
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1answer
104 views

Centre of $GL(n,\mathbb{R})$ [duplicate]

Can you help me regarding the Centre of $GL(n,\mathbb{R})$ $?$ It is easy to see that the diagonals are there. what could be the other elements? It may be an useless question but it came to my mind! ...
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1answer
62 views

Residually Finite group $\Rightarrow$ Totally disconnected

How can I prove that a residually finite group $G$ is totally disconnected? I considered the topology generatad by $\{Ng\}_{N\in\eta,\;g\in G}$ where $\eta=\{N\unlhd G \;, |G:N|<\infty\}$ and I ...
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1answer
58 views

Simple question on topological groups

Why is $\{1\}$ closed in a totally disconnected topological group?