The study of symmetry: groups, subgroups, homomorphisms, group actions.

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Counting of edges coloring in a graph

The problem is to count of coloring graphs. We have three colors. And I found all automorphisms. It is: $$\alpha_1: (1)(2)(3)(4)(5)(6)$$ $$\alpha_2: (123456) $$ $$\alpha_3: (135)(246) $$ ...
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2answers
20 views

Isomorphism between quotient groups

Exercise Let $f:G \to G'$ be an isomorphism and let $H\unlhd G$. If $H'=f(H)$, prove that $G/H \cong G'/H'$. As I've shown that $H'\unlhd G'$, I thought of defining $$\phi(Ha)=H'f(a)$$I was trying ...
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0answers
49 views

On the definition of groups of multiplicative type

Let $k$ be a field of characteristic 0. The definition of a linear algebraic $k$-group of multiplicative type (m.t.) I've seen the most in the literature is that $G$ is of m.t. if it is a ...
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1answer
73 views

How do we know the classification of finite simple groups is finished?

I don't have much experience in group theory beyond knowing what a finite simple group is, as well as the other basic parts of the topic, so far as I can tell, so if the actual answer is too complex ...
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42 views

For $G$ a group of order $1925$, find the number of Sylow $5$-subgroups

Let $G$ be a finite group of order $1925$. Find the number of Sylow $5$-subgroups in $G$. There must be $1$ or $11$ such subgroups. What is the actual number?
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2answers
51 views

No. of homomorphisms from $\mathbb Z_n$ to $\mathbb Q$

How many homomorphisms are there from $\mathbb Z_n$ to $\mathbb Q$ ?
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1answer
24 views

Elements in the same coset and the Cayley Diagram

A question from Visual group theory, by Nathan Carter. In a Cayley diagram, if $aH$ is a coset of a subgroup $H$ of a group $G$ and $b$ belongs to $aH$, why is it that every node that can be reached ...
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1answer
47 views

Constructing a group automorphism making a diagram commute

Let $A,B$ be groups, suppose we have an epimorphism $p:A \to B$. Let $\phi \in \operatorname{Aut}(A)$. Does there exist some $\varphi \in \operatorname{Aut}(B)$ such that the following diagram ...
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37 views

The number of distinct conjugates of a $p$-subgroup

Let $G$ be a $p$-group and $H$ be a $p$-subgroup of $G$; $H$ is not normal in $G$. Then prove that the number of distinct conjugates of $H$ in $G$ divides $|G|$. Notice that the conjugates may have ...
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1answer
54 views

Order of group $GL_{2}\left( \mathbb{F}_{p}\right) $

I'm having a hard time counting. I need to count the number of elements for the multiplicative group of invertible $2\times 2$ matrices $GL_{2}\left( \mathbb{F}_{p}\right) $ with elements from the ...
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1answer
64 views

Proving Finiteness of Group from Presentation

Given the group $G = \langle a, b, c : a^2 = b^3 = c^5 = abc\rangle$, I want to show that $H = G / \langle abc\rangle$ is a finite group. I tried to find a canonical form for elements of $H$. That ...
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1answer
35 views

Problem on normal subgroups

Problem Let $G$ be a group and $H,K$ subgroups of $G$, we define $HK=\{h.k : h \in H, k \in K\}$. Prove that if $H$ or $K$ is normal, then $HK$ is a subgroup. In order to show $HK$ is a subgroup, ...
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1answer
192 views

Is true that : group of exponent 4 implies that $[[[x,y],y],y]= \text{identity}$?

It is well known that: If the square of every element of a group is the identity then the group is abelian. Also is known that: In a group, if (for all $x$) the cube of $x$ is the identity (i.e. a ...
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2answers
41 views

Abelian group and morphism equivalent statement

Exercise Show that the following statements are equivalent: $(i) \space G \space \text{is abelian.}$ $(ii) \space \text{the map f: G} \to \text{G defined as} \space f(x)=x^{-1} \space \text{is a ...
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3answers
140 views

Number of Automorphisms on group

I had a doubt in the following question How many automorphisms possible on the group $Z^{+}_{12} $. Although I believe it should be $12!$ but want to confirm. Thanks
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3answers
49 views

When is a non-trivial homomorphism injective?

I noticed that over the natural numers $(\mathbb{Z},+)$ any group homomorphism $f : \mathbb{Z} \rightarrow \mathbb{Z}$ that is not the trivial one, is automatically injective. Where exactly does ...
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56 views

Let $G$ be a group and $K$ be subgroup of order $p^a$. I am trying to show that $|\{H \leq G : K \subset H, |H|=p^{a+b}\}| \equiv 1 \pmod{p}$

Let $G$ be a group and $K$ be subgroup of order $p^a$. I am trying to show that $$|\{H \leq G : K \subset H, |H|=p^{a+b}\}| \equiv 1 \pmod{p}$$ for each b fixed & for all $a+b$ such that ...
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1answer
28 views

Iterated wreath product

Can someone tell me what is iterated wreath product or where I can appropriate definition? I'm trying to understand one paper and author claims that some elements $g_1,...,g_k$ are each of order ...
4
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1answer
57 views

To show that a concretely defined group is isomorphic to an explicitly presented group, what strategies are available?

I have a homework problem of the following form. We're given presentation of a group $\langle x,y \mid R\rangle$ explicitly, and two matrices $X,Y \in \mathrm{GL}(\mathbb{C},2).$ We know $X$ and $Y$ ...
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2answers
24 views

Does the following binary operation form a group on a set with 3 elements? (multiple identities?)

Let S = {a, b, c}. *| a b c ----------- a| a b c b| b a a c| c a a This seems to have all the desired characteristics of a group, however, both b and c ...
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1answer
45 views

How to determine the isomorphism types of given groups with generators and relations

I was classifying the all groups of order 30 and I got the following groups $\langle a,b \mid b^{-1}ab=a^4, a^{15}=b^{2}=1\rangle$ and $\langle a,b \mid b^{-1}ab=a^{11}, a^{15}=b^{2}=1\rangle$. How ...
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2answers
23 views

A cycle as a product of transpositions

Can someone please explain how a cycle $(1234)$ can be written as a product of transpositions: $(14)(13)(12)$? And how they can be multiplied to (1234)? Thanks in advance.
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1answer
37 views

The order of an element of a group $G$

Exercise Let $G$ be any group and $x=a^k$, where $a \in G$ is an element of order $n$ and $k$ is a natural number. Find the $ord(x)$. My candidate for $ord(a^k)$ is $\dfrac{n}{(n:k)}$. By elevating ...
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188 views

Show that the kernel of the map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion.

I am trying to show that the kernel of the natural map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion. That is, if $A$ is in the kernel then $A = I$ or $A^n \neq I$ for all $n ...
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1answer
42 views

Cauchy's theorem converse

Cauchy's theorem states that if p is a prime dividing the order of a group G, then there is an element of order p. how about if p is a prime the order of an element of a group G, p always divide the ...
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23 views

How to reconstruct geometric object that a Frobenius group acts on

A Frobenius group has equivalent definitions: a transitive permutation group on a finite set such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. ...
2
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1answer
21 views

Explicit formula for invariant inner product of the standard representation of $S_3$

Let $V$ be a representation of a group $G$ over $\mathbb{C}$. Given the standard Hermmitian inner product $\langle\cdot,\cdot\rangle$ on $V$ we can always define a $G$-invariant inner product by ...
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Prove identity involving alternating groups

Prove the following identity: where $I_{A_n}(x_1,...,x_n)$ is a cyclic index the natural action of the alternating group $A_n$ on the set ${1,...,N}$ (assuming that $I_{A_0} = 1$).
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Union of subgroups is subgroup

I am doing an exercise where I am asked to prove or disprove the statement: If $G$ is a group, and $H_1,H_2,H_3$ and $H_1 \cup H_2 \cup H_3$ are subgroups of $G$ then $\exists i,j$ with $i \neq j$ ...
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1answer
19 views

Is $\{f \in End_{\mathbb R}(\mathbb R^n) : d(f(x),f(y))=(x,y) \space \forall x,y \in \mathbb R^n\}$ a group?

I am trying to figure out if the set $\{f \in End_{\mathbb R}(\mathbb R^n) : d(f(x),f(y))=(x,y) \space \forall x,y \in \mathbb R^n\}$ is a group under the operation composition ($d$ is a metric). ...
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1answer
40 views

Polycyclic groups and Group extension

Suppose we have a SES $1\to N\to G\to G/N\to 1$, and assume that $G/N$ is polycyclic. What condition on $N$ will ensure that $G$ is polycyclic? THanks.
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1answer
37 views

Characterizing groups with linear subgroups

Is there a simple characterization for a group $G$ satisfying $$(\forall H,K\le G)(H\subseteq K\text{ or } K\subseteq H)$$
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35 views

When can we find coset representatives that generate the original group?

Let $G$ denote a group and consider a subgroup $H \subseteq G$. Then the left cosets of $H$ form a partitioning of $G$ (irrespective of whether or not $H$ is normal). Call this partitioning $\Pi_H$. ...
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Both elements are identity [duplicate]

In a group $G$, suppose there are elements $a,b\in G$ satisfying $$ a^{-1}b^2a=b^3\quad \text{ and }\quad b^{-1}a^2b=a^3$$ How to show that $a=b=\rm e$. Where $\rm e$ is the identity element of $G$. ...
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1answer
29 views

Order of a group and cyclic group theory connection

Just stuck on a problem. If group $G$ of order $6$ contains an element of order $6$, then prove that $G$ is a cyclic group of order $6$. Any hint will be appreciated.
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4answers
93 views

An infinite group $G$ and $\forall x\in G, x^n=e$

Let $G$ be an infinite group and $n\in \mathbb N$. If for any infinite subset $A$ of $G$ there is $a\in A$ such that $$a^n=e,~~~~(e=e_G)$$ then prove that for every element $x\in G$ we have ...
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1answer
37 views

There is no group whose quotient by the center is isomorphic to the quaternion group [duplicate]

I have to prove that there is no group $G$ such that $G/Z(G)$ is isomorphic to $Q_8$. Anytone can give me an idea to begin? thanks
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2answers
70 views

Groups, inverse Galois problem and transcendence degrees

This is a curiosity of mine. I suspect there might be a trivial answer, but if there is none, this problem will probably haunt me for a long time... The question is as follows : Given a group $G$, ...
2
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1answer
53 views

$GL_2(R)$ and $\mathbb{A}_4$

So, i have to prove that $\mathbb{A}_4$ is not isomorphic to a subgroup $G$ of $GL_2(R)$. Here is a "hint" (are previuos part of the same problem that i was thinking, so i supose should help): If ...
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3answers
130 views

Show that $\mathbb Z[x]$ and $\mathbb Q_{>0}$ are isomorphic [closed]

Let $(\mathbb{Z}[x],+)$ be the additive group of all polynomials with integer coefficients and $ (\mathbb{Q}_{>0},*)$ the multiplicative group of all positive rationals. (Please) Show (me) these ...
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1answer
73 views

Field extension $\mathbb Q(f)/\mathbb Q$ and its Galois group

Let $E/\mathbb{Q}$ and $F/E$ be finite extensions of fields, let $u$ be an element of $Aut(F/E)$, and let $f$ be an element of $F$. Suppose that (i) $[F:E]=3$, (ii) $F=\mathbb{Q}(f)$ and ...
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1answer
44 views

The Galois group of a polynomial

I was just looking for some clarification regarding the definition of the Galois group of a polynomial $f(x)$. So, if I remember correctly, this is defined as the Galois group of a splitting field of ...
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38 views

Figuring out automorphism groups

I was wondering what tactics people usually use to figure out automorphism groups. Let's start with small finite groups. For example, I'm trying to figure out $\mathrm{Aut}(V_4)$. My thought process ...
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1answer
24 views

The number maximal subgroups of a 2-generated group

Let $G$ be a 2-generated group. Then prove that the number subgroups of index 2 is at most 3. By Hint i think we have at most 3 cases: Let $G=\langle a,b\rangle$ and $C_{2}=\langle x\rangle$. Then ...
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104 views

About Commutators in Subgroups

Let $G$ be a group and $H$ a subgroup of $G$. Is clear that if $x$ and $y$ are elements in $H$ then $[x,y] = x^{-1}y^{-1}xy \in H$. But, is true that, if $1 \neq [x,y] \in H$, then $x$ and $y$ are ...
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30 views

Show that $\mathbb{F}_p^\times \simeq \text{Aut}(\mathbb{F}_p^+) $ holds

I want to show, that $\mathbb{F}_p^\times \simeq \text{Aut}(\mathbb{F}_p^+)$ holds with $p$ prime. ($\mathbb{F}_p^+$ is the additive group, $\mathbb{F}_p^\times$ multiplicative group) As hint we ...
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1answer
33 views

Proof of the theorem that relates the size of the conjugacy class to the order of the centralizer subgroup

Could you please explain the step in this theorem and proof that I do not follow: If $G$ is a finite group, let $C_{G}(x)$ be the centralizer of $x$ in $G$, that is $C_{G}(x)=\{g \in G: xg=gx\}$ ...
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52 views

Prove that $\operatorname{Aut}(S_n)$ isomorphic to $S_n$ when $n\geq 3$ and $n \neq6$.

Prove that $\operatorname{Aut}(S_n)$ isomorphic to $S_n$ when $n\geq 3$, and $n \neq 6$. I can see that the automorphisms of $S_n$ have the same structure as $S_n$. But I am having trouble ...
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0answers
41 views

Prove this group $G$ is abelian [duplicate]

Let $G$ be a finite group and $\alpha$ be an automorphism of $G$ which fixes only the unit of $G$ (if $\alpha(a)=a$, then $a=1$). And $\alpha^2=1$. Show that $G$ is abelian. I think it is enough ...
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1answer
51 views

How prove this $Z(H)\neq 1$, if for any $g\in G\setminus H, H\cap H^g=1$

Let $2||H|$, and let $H$ be a subgroup of $G$, $H\le G$, such that for any $g\in G\setminus H$ the following holds. $$H\cap H^g=1$$ Show that :$$Z(H)\neq 1$$ where $Z(H)$ is center of the $H$. ...