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

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A subgroup of a cyclic normal subgroup of a Group is Normal

Is it true that Subgroup of a Cyclic Normal subgroup of a Group is again Normal ? If so any hints for the proof?
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192 views

Group of sphere transformations, impressing friends

Ok, so here's the story: I am reading a book on algebra and, via some exercises, discovered that in any group $G$, the order of $x \cdot y$, written $o(x \cdot y)$, equals $o(y \cdot x)$. Now, this is ...
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70 views

normal subgroup + Socle

I am looking for examples of socle and normal-subgroup relations. If $G=S_{4}$, the normal subgroups are $A_{4}$ and $V_{4}$, thus the socle of $G$ is the klein-four group, and for $A_{4} \cap Soc(G) ...
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1answer
133 views

Maximal subgroups of almost simple groups with socle $PSL(2, q)$

Let $G$ be an almost simple group with socle $PSL(2,q)$ where $q=p^f>3$ is the $f$th power of some odd prime $p$, and $M$ a maximal subgroup of $G$. By http://arxiv.org/abs/math/0703685, except for ...
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456 views

What's the difference between the center of a group and a normal subgroup?

It seems the definition of the center of a group and a normal subgroup are the same so I'm wondering what the difference is between the two? A group $H$ is normal in $G$ iff $Hg=gH$ for all $g \in ...
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33 views

Constructing a group with subsets

Given the set $S=\{0,1,2\}$, it has been asked to prove that it is not a group under the operation $\max(x,y)$. It can be done. Then they ask to identify $3$ subsets of $S$ which are 'groups' under ...
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1answer
296 views

If an abelian group G of order 10 contains an element of order 5,how can i prove that G must be a cyclic group [duplicate]

Possible Duplicate: Prove that G is a cyclic group If an abelian group $G$ of order $10$ contains an element of order $5$ ,how can I prove that $G$ must be a cyclic group. i am completely ...
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95 views

A step in the proof of Cauchy's theorem for groups

Let $G$ a group, $p$ a prime and $X=G^p$. Let $\sigma\in S_X$ act as follows: $\sigma(x_1,...,x_p) = (x_2,...,x_p,x_1)$. Let $Y$ the subset of elements in $X$ such that $x_1x_2...x_p=1$ and let ...
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1answer
121 views

Injective Homomorphism on $S_n$

I have the following question on a homomorphism between symmetric groups and it has been really a pain. I know I am supposed to use induction but, I seem to miss something essential so if anybody ...
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1answer
254 views

Image of conjugacy class under surjective homomorphism

There is a surjective homomorphism from $G$ to $G'$. Let $C$ denote the conjugacy class of element $x$ in $G$, $C'$ the conjugacy class of the image of $x$ in $G'$. Prove the order of $C'$ divides the ...
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140 views

Regular subgroups of affine general linear groups

Let $AGL(2d,2)$ be the affine general linear group acting natrually on a $2d$-dimensional vector space over $GF(2)$. Is there a regular subgroup of $AGL(2d,2)$ isomorphic to $Z_{2^d}:Z_{2^d}$ for ...
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89 views

Product of subgroups and generating sets

Prove or disprove the following: $(1)$ Let $H$,$K$ be subgroups of a finite group $G$. If $|HK|=|\langle H,K \rangle|$, then $HK=\langle H,K \rangle$, that is, $HK$ is a subgroup of $G$. $(2)$ Let ...
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83 views

Isomorphic representations on exterior powers

Exercise from F+H, Exercise 1.3: Let $\rho : G \rightarrow GL(V)$ be any representation of the finite group $G$ on a $n$-dimensional vector space $V$ and suppose that for any $g \in G$ the ...
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203 views

Showing a composition is isomorphic to additive group

Consider the law of composition $(x,y) \rightarrow \sqrt[3]{x^3+y^3}$ on the set of real numbers. Show that this law of composition defines a group isomorphic to the additive group ($G$) of ...
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1answer
168 views

Normality in Todd-Coxeter Algorithm?

How is normality reflected in Todd-Coxeter Algorithm tables? I can think of the generic cases like if there are only 2 indices then the subgroup has index 2 and thus is normal. Same case for p ...
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30 views

$(g,k)$ modules for $\mathrm{SU}(1,1)$

Let the group be $\mathrm{SU}(1,1)$, choose maximal compact subgroup $$ K_{\mathbb{R}}=\left\{ \left(\begin{array}{cc} e^{i\theta} & 0\\ 0 & e^{i\theta} \end{array}\right),\,\theta\in ...
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2answers
201 views

Group generated by $x$,$y$|$x^2=y^2=1$

Classify groups generated by two elements $x$, $y$ of order 2. First, please help clarify: Is it necessary for $x$ and $y$ to be distinct elements? (above is the whole question as seen in the book). ...
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107 views

Hall subgroup of a finite group?

How did the author get that $L=(L \cap H)(L\cap K)$ in Lemma $5$ below. Remark: All the groups here are finite. $H$ permutes (commutes) with $K$ means $HK=KH$ where $H$ and $K$ are subgroups of some ...
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2answers
160 views

$G=\langle a,b\mid aba=b^2,bab=a^2\rangle$ is not metabelian of order $24$

This is my self-study exercise: Let $G=\langle a,b\mid aba=b^2,bab=a^2\rangle$. Show that $G$ is not metabelian. I know; I have to show that $G'$ is not an abelian subgroup. The index of $G'$ ...
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64 views

Soluble subgroups of finite classical simple groups

What do we know about the soluble subgroups of finite classical simple groups (and maybe their automorphism groups)? For example, are their maximal soluble subgroups explicitly known?
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144 views

Greatest elements in crystallographic root systems

I have a question regarding a remark in the book "Reflection Groups And Coxeter Groups" by James E. Humphreys (unfortunately the book is not to be found as a whole on google books or such). In ...
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1answer
96 views

Find a Group with following presentation

Can anyone describe a group with following presentations? (rigorous proof is not needed) $$ \langle x,y,z \mid x^2, y^2, z^2, (xz)^2, (xy)^3, (yz)^3\rangle $$
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1answer
60 views

Name for a member in a group that can cover all members by all of its exponentials?

In a group $G$, if there is a member $x$ of $G$ s.t. $G=\{x^n, n \in \mathbb{N} \text{ or } \mathbb{Z}\}$, is there a name for such $x$? Thanks!
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155 views

Group(Non-abelian) Multiplication Table

Given Group(Non-abelian) Multiplication Table. Find $ca,bb\text{ and }df$. $$\begin{array}{ l | c r r r r r } * & e & a & b & c & d &f \\ \hline e & e & ...
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3answers
185 views

Order of a group element. [$\operatorname{ord}(x)$]

$\newcommand{\ord}{\operatorname{ord}}$ I need to prove the following $(i)$ $\forall x\in G\,:\ord(x)=\ord(x^{-1})$ $(ii)$ $\forall x,g\in G\,:\ord(gxg^{-1})=\ord(x)\qquad$ (for ...
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331 views

Clarification of McKay's Proof of Cauchy's Theorem for Groups

In an earlier post, I asked for some intuition on Cauchy's Theorem for Groups (for every prime divisor of the order of a finite group, there exists an elements who's order is that prime). I got great ...
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162 views

$G'$ and the product of all the elements in an odd order group

Yesterday, through working on a question Groups with only one element of order 2, Don antonio brought out a nice question within the comments: The product of all the elements in an odd order group ...
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1answer
248 views

Proof about cosets

C In parts 1-5 below, $G$ is a group and $H$ is a normal subgroup of $G$. Prove the following (Theorem 5 will play a crucial role) Theorem 5- Let G be a group and H be a subgroup of G. Then $(i)$ ...
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Question on Abelian Group

Let $G$ be an abelian group. Show that $\{x\in{G} | |x| < \infty\}$ is a subgroup of $G$. Give an example of a non-abelian group where this fails to be a subgroup.
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60 views

How many semi-direct products $H \rtimes Q$ can be be constructed for $H \cong C_{42}, Q \cong C_{3}$

I was also given a hint: Represent $C_{42} \cong C_2 \times C_3 \times C_7$, find its group of automorphisms, then look for elements of order 3. So, I found the group of automorphisms to be: $$ ...
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104 views

What is an example of a situation where AB is not subgroup of G, when A, B are subgroups of G?

What is an example of a situation where AB is not subgroup of G, when A, B are subgroups of G? My first instinct is always to go for some dihedral group or other...But I could not find an example of ...
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216 views

$f\colon I\rightarrow G$ and Gromov $\delta$-hyperbolicity

Please recall that $\left|\int_0^1 f(t)\,dt -w\right|\leq \int_0^1|f(t)-w|\,dt$. In general, let $(X,d)$ be a metric space. Given a function $f:I\to X$ let $m_f\in X$ be such that $d(m_f,w)\leq ...
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27 views

Degrees of parabolic subgroups

Suppose a finite reflection group $G$ has the degrees $d_1,\ldots,d_n$. Let $G^*$ be a parabolic subgroup of $G$. What are the degrees of $G^*$. Since $|G^*|$ divides $|G|$ it is clear that the ...
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3answers
615 views

A question regarding the definition of Galois group

In my book, Galois group is defined to mean the set of automorphisms on $E/F$ that "leave alone" the elements in $F$. On Wikipedia it says: "If $E/F$ is a Galois extension, then $Aut(E/F)$ is called ...
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1answer
62 views

Proving normality of a subgroup under componentwise multiplication

The question: Let $G\times H $= {$(g,h) : g \in G, h\in H$}, where G and H are groups. Let $G_1$ = {$(g,e_H): g \in G$}. Prove that $G_1$ is a normal subgroup of $G\times H$ and that G is isomporphic ...
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130 views

A cyclic group of order “rs” where (r, s) = 1

I was given this question and I'm not really sure how to approach this... Assume $(r,s) = 1$. Prove that If $G = \langle x\rangle$ has order $rs$, then $x = yz$, where $y$ has order $r$, $z$ has ...
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3answers
116 views

Showing homomorphism for $\theta: GL_2 (\Bbb Q) \rightarrow \Bbb Q\setminus \{0\}$ given by $\theta(A) = \det A$.

Show that this map is a group homomorphism and find its kernel: $$\theta: GL_2 (\Bbb Q) \rightarrow \Bbb Q\setminus \{0\}$$ given by $\theta(A) = \det A.$ My attempt: Let $A = \begin{pmatrix} ...
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1answer
252 views

Show that each of the following are homomorphisms

Show that the following maps are group homomorphisms and find their kernels: 1) $\theta: \Bbb Z \rightarrow GL_2$ $\theta(n) = $$ \begin{pmatrix} 1 & n \\ 0 & 1 \\ ...
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0answers
78 views

Asking about Ulm invariant of abelian group $G$

I am backing to read my previous question and learn more facts which the Masters left for me within comments. One of them appeared here Verifing some properties about $G$. There; I was verifying that ...
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2answers
41 views

Having trouble with a problem involving factor groups.

I'm having a lot of trouble with this problem. If anyone could help me out it would be much appreciated: Suppose $ G $ is a group and $|G : Z(G)|=4$. Prove that $G/Z(G)\approx \mathbb Z_2 \oplus ...
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60 views

A condition involving internal direct products.

We were given the following statement to prove: "in $ \mathbb Z $, let $ H= \langle 3 \rangle $ and $ K = \langle 7 \rangle $. Prove that $ \mathbb Z = HK $. Does $ \mathbb Z = H \times K $?" I have ...
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showing that $\frac{\mathbb R}{\mathbb Q}\cong\mathbb R$, both over $\mathbb Q$

In my last question The following groups are the same., three groups were given and I wanted to verify that they are isomorphic to each other. Derek suggested some points and I got my answer about one ...
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1answer
199 views

find presentation of semidirect product (C2 x C2) ⋊ C2

I would really appreciate a step by step about how to solve the following: C2 rwr C2 (where rwr is the regular wreath product) i know it becomes (C2 x C2) ⋊ C2, so how to go from here. edit: with ...
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1answer
30 views

How to prove that there are at least two different unital homomorphisms for field $K\rightarrow K$

How to prove that exists a field K such that there are two unital homomorphisms between fields $f:K\rightarrow K$? Homomorphism is unital if $f(1) = 1$
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2answers
129 views

$T(G)$ may not be a subgroup?

It is obvious that for an abelian group $G$; the set of all torsion elements: $$T(G)=\{x\in G|x^n=0 \text{, for some nonzero integer } n\}$$ is a subgroup of the group. I am asked to probe this fact ...
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173 views

Cauchy's Theorem for Groups

Specifically: If $p$ is a prime divisor of the order of a finite group $G$, then there exists an element of order $p$ in $G$ So I'm looking for a little intuition behind this idea. I understand how ...
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1answer
58 views

Describing a homomorphism $f : GL_n (\mathbb{R}) \rightarrow (\mathbb{R^*}, \cdot)$

Let $f : GL_n (\mathbb{R}) \rightarrow (\mathbb{R^*}, \cdot)$ be a group homomorphism such that $\forall a \in G$ $$ f(a) = det(a) $$ (a) Describe $Ker(f)$ (b) Describe $Im(f)$ ...
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208 views

Abelian Groups of order 2000

Classify, up to isomorphism, all abelian groups of order 2,000, giving the standard form of each group in your list. (The standard form is also called the invariant factor decomposition.)
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860 views

Was my abstract algebra textbook trying to kill me?

I'm doing an independent study (self-taught course) in abstract algebra and I'm using the abstract algebra textbook here: http://abstract.ups.edu/. In Chapter 9: Isomorphisms, problem 20 asks: "Prove ...
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1answer
118 views