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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Decomposition $\mathbb{Z}_{p^n}\cong G_1 \times G_2$ for $p$ prime

I want to prove that if $p$ is a prime and $\mathbb{Z}_{p^n}\cong G_1 \times G_2$, then either $G_1 \cong \mathbb{Z}_{p^n}$ or $G_2 \cong \mathbb{Z}_{p^n}$ I have tried to approach this problem with ...
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22 views

Restricting monomorphisms to isomorphisms

Suppose we have a group/ring monomorphism $\phi: R\to S$. Then we have an isomorphism $\phi|_S :R\to \phi(S)$ So now we can find inverse of $\phi|_S=\phi|_S^{-1}: \phi(S) \to R$ irrespective of ...
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154 views

Group of integer orthogonal matrices

Let $O_n(\mathbb Z)$ be a group of orthogonal matrices B st. $B*B^T=I$ with entries $b_{ij} \in \mathbb Z$. How do I show that $O_n(\mathbb Z)$ is a finite group and find its order. I need to show ...
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44 views

Is the relative product in the category of Abelian groups the same as the relative product in the category of Groups?

I'm not sure if "relative product" is common terminology, but given $N \xrightarrow{f} A \xleftarrow{g} K$ in a category, the relative product is the diagram $$\begin{array} A N\times_A K & ...
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Group cohomology: $\mathbb{Z}G$-maps from $G^{n+1}$ are the same as set maps from $G^n$

I'm learning about group cohomology from Knapp's book Advanced Algebra. Given a group $G$, and an abelian group $M$, he defines $F_n$ to be the $(n+1)$-fold product of $G$, endowed with a $G$-action ...
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45 views

Sylow subgroups induce Sylow subgroups in normal subgroups

Let $P \in Syl_p(G)$ and assume N is normal in G. Use conjugacy part of sylow's theorem to prove that $P \cap N$ is a sylow p-subgroup of N. Deduce that PN/N is a sylow p-subgroup of G/N. How do we ...
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28 views

Index of intersection of finite index subgroups

I'm trying to understand this proof http://math.stackexchange.com/a/128549/205193 which does not seem complicated but I don't understand why : why $p(x)=p(y)$, implies that $x$ and $y$ are in the ...
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91 views

Proof that $A_n$ is simple for $n \ge 5$, is the one presented here overcomplicated?

In the book Permutation Groups by Dixon & Mortimer, page 78, the well-known fact that for $|\Omega| \ge 5$ the alternating group $Alt(\Omega)$ is simple is proven. It uses a Theorem that if a ...
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38 views

Finite dimensional representation of infinite group cannot be unitary: example with $\mathbb R$

Consider the representation of the group of real numbers $\mathbb R$ given by $$ \rho (x) = \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix} $$ for $x \in \mathbb R$. How can we see that this ...
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45 views

Show a group is a semi direct product

Consider a set $G$ of $n \times n$ matrices with entries $\{0,1,-1\}$ that have exactly one non zero entry in each row a column. Show $G$ is a group and that $G$ is the semi direct product of the ...
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34 views

Classification of symmetric space of non compact type

Is there a classification of rank one symmetric space of non compact type ? Remark, for rank 1, There are three families : 1) hyperbolic n-space, corresponding to the Lie group SO(n,1). 2) complex ...
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28 views

What are the inverses of this group?

Consider the group $G= \mathbb{Z}^{*}_{7}$ under multiplication. What are the inverses? I'm guessing that the identity element is 1, but then how can any $x \in G$ such that $x>1$ have an inverse? ...
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122 views

Coset Enumeration with GAP

I have downloaded GAP version 4.7.8. for windows, and installed everything (all packages, including"ACE") with the installer. Now I want to do a simple task, enumerating the cosets. To create a group ...
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53 views

A (continuous) inverse to derive (commutator sub group) functor on the Category of Groups

Is there a functor $\mathcal{F}$ on the category of groups which satisfy $$D \circ \mathcal{F}=\mathrm{Id}$$ where $D$ is the derive (commutator subgroup) functor? Is this (possible) ...
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56 views

if $H$ is a unique subgroup of order $d$ then $H$ is a normal subgroup

Let $H$ be a subgroup of a group, $G$ such that $H$ is the only subgroup in $G$ with order $d$. Prove that $H\vartriangleleft G$. I have to questions: 1) as you see it is not given that $G$ is ...
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24 views

If $G$ is any infinite cyclic group then how do you prove that $\operatorname{Aut}(G)=\{e_G, -e_G\}$

What I've done so far: Say $G$ is generated by some fixed $a$ and $f \in\operatorname{Aut}(G)$. Then suppose $f(a) = \alpha \in G$. How do I go from here to show that $f(x)=x$ or $f(x)=x^{-1}$ for ...
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23 views

How do I find a subgroup $H$ of $S_3$ that is isomorphic with $G=A_3$ using the group left action method?

I don't understand what I'm to do here very well. I started with defining the elements of $G$ as $e=$ identity, $x=(1 \, 2 \, 3)$ $y=(1 \, 3 \,2 )$ then left multiplying each element of $G$ ...
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77 views

group of rigid motions dodecahedron

We have the following theorem If |G| = 60 and G has more than one Sylow-5 subgroup, then G is simple. Since order of the rigid motion of the dodecahedron group is 60, so all we have to do is to ...
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Action of a finite abelian group on a finite set

Let $G$ be a finite abelian group acting on a finite set $X$. Let $\mathcal{O}(x)$ be the orbit of $x$: $$\mathcal{O}(x)=\{g.x\colon g\in G\}.$$ Then obviously $G$ acts transitively on ...
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36 views

Direct product of free groups as quotient of free product

Let $F(X)$, $F(Y)$ be free groups, generated by (finite) sets $X$ and $Y$. Is there a good description of kernel of $f:F(X)*F(Y)\to F(X)\times F(Y)$, where $f(w_1\dots w_n)=(w_1,\dots,w_n)$ ? All I ...
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107 views

Meaning of natural homomorphism

When I reached the homomorphism concept in group theory, I thought that "natural" is only an ordinary literal word without mathematical meaninig. But I read somewhere this word has a precise ...
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34 views

Why is $\operatorname{Hom}_{\textrm{AG}}(k^\times,k^\times)\cong\mathbb{Z}$?

Suppose $k$ is an algebraically closed field, and $k^\times$ its multiplicative group. I read that $\operatorname{Hom}_{\textrm{AG}}(k^\times,k^\times)\cong\mathbb{Z}$, where the left consists of ...
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I don't know what this symbol means

I somehow made it to grad school without coming across this symbol: $\left( \begin{array}{ccc} l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3 \end{array}\right) $ Here, $l_i$ and $m_i$ are all ...
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40 views

Let $G$ be a group such that $(ab)^3=a^3b^3$ for all $a, b\in G$. Prove that $H=\{x^6\mid x \in G\}$ is a subgroup of $G$

Let $G$ be a group such that $(ab)^3=a^3b^3$ for all $a, b\in G$. Prove that $H=\{x^6\mid x \in G\}$ is a subgroup of $G$ Attempt: $e=e^6$ i.e $e \in H\neq \phi$ Let $a,b\in H$ then $a=x^6, ...
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89 views

Let $G$ be a p-group with $|G|=p^n$. Show $G$ has normal subgroups of order $p^m$ for each $0<m<n$.

Here is what I have so far. Statement: Let $G$ be a p-group with $|G|=p^n$. Then $G$ has normal subgroups of order $p^m$ for each $0<m<n$. The case $|G|=p^1$ is trivial. Proceed by induction ...
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Show $ M $ is a characteristic subgroup of $ K $

Let $ G $ is a soluble group. Let $ K/L $ be a chief factor of $ G $. Then $ K/L $ is a minimal normal subgroup of $ G/L $. Suppose $ M $ is the smallest normal subgroup of $ K $ that $ K/M $ is ...
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If $k|n$, then $D_{2n}/\langle{r^k}\rangle\cong D_{2k}$

I have to show that, given $k|n,$ and $\langle{r^k}\rangle$ is a normal subgroup of the Dihedral group $D_{2n}$ then $D_{2n}/\langle{r^k}\rangle\cong D_{2k}$ Given this, I know that I need to show ...
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77 views

Are the fibers of a group homomorphism the cosets of its kernel?

Given a group $G$ and a homomorphism $f : G \to G'$, $f$ induces a relation on $G$, which we will denote $\sim$. The relation is $g_1\sim g_1$ iff $f(g_1) = f(g_2)$. The fiber of an element $y \in G'$ ...
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30 views

problem with showing some relations in finite non abelian p-group

In some paper we have I don't get last two line, I don't know how to prove any of them. if you can give me some hint it would be good $\Omega_1=$ subgroup generated by all elements of order p ...
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58 views

A special subgroup of $S_{10}$ generated by four involutions

Is there an easy way to describe the subgroup of $S_{10}$ generated by the four involutions: \begin{align*} & (4 ~ 7)(5 ~ 8)(6 ~ 9) \\ & (2 ~ 7)(3 ~ 8)(6 ~ 10) \\ & (1 ~ 7)(3 ~ 9)(5 ~ ...
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33 views

Show that $o(gHg^{-1})=o(H)$

Let $H$ be a subgroup of $G$. Prove that $gHg^{-1}=\{ghg^{-1}|h \in H\}$ is a subgroup of $G$, $g\in G$. Show that $o(gHg^{-1})=o(H)$. Attempt: As $e=geg^{-1}\in gHg^{-1}$ then $gHg^{-1}$ is ...
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Structure of the cosets of the finite group

Suppose $G$ is a finite group and $H,K$ are subgroups such that $$K < H < G.$$ Is there a relation between $H$-cosets and $K$-cosets? Thank you for your suggestions.
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48 views

Number of Involutions generating Symmetric Group

How to show that strictly less than $n-1$ involutions (of which the transpositions are a special case) could not generate $S_{n}$ for $n > 3$? I know that $n-1$ transpositions are sufficient, or ...
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41 views

Sub-group of the modulo group is an ideal

how do I show that every sub-group of ring $Z_n$ is an ideal in $Z_n$? If $n$ is prime, the only sub-groups are the trivial and that mean's they are ideals, but if $n$ isn't prime, there are non ...
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Intersection of Normal Subgroups is normal, set approach

Assuming that we know that, given two normal subgroups $H,K$ of a group $G$ that their intersection is also a subgroup of $G$, the goal is to show that $H\cap K$ is also normal. I saw a couple of ...
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1answer
45 views

Let $G$ Be a group and $K\subset H \subset G$. $H$ be a subgroup of $G$, $K$ be a subgroup of $H$. Prove that $K$ is a subgroup of $G$

Let $G$ Be a group and $K\subset H \subset G$. $H$ be a subgroup of $G$, $K$ be a subgroup of $H$. Prove that $K$ is a subgroup of $G$. Assume that operations are always the same. Attempt: Let ...
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Proving Lagranges Thorem

Im trying to understand Lagranges theorem.There is this group G and subgroup H. I have the right coset of H (order m) I have this statement in the proof Let which is impossible What do they ...
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Finding the linear mapping between two equivalent irreducible representations

Suppose $V_1$ and $V_2$ are two finite dimensional linear vector spaces providing two equivalent irreducible representations of a group $G$. We know that there exists a linear map $T: V_1 ...
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Show by Lagrange's theorem that a group $G$ of order $27$ should have a subgroup of order $3$.

Show by Lagrange's theorem that a group $G$ of order $27$ should have a subgroup of order $3$. Attempt We have $o(a)|o(G)$ for all $a\in G$. Then the possibilities of orders for the elements og ...
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if $a^{-1}ba=b^i$ then $a^nba^{-n}=b^{(i^n)}$ for all $n$

Let $a$ and $b$ be to elements in a group such that $a^{-1}ba=b^i$ for some natural $i$. prove that $a^nba^{-n}=b^{(i^n)}$ for all $n$. I tried to manipulate the given expression but I didn't manage ...
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45 views

Order of elements in group and in permutation representation of that group

If $G \le Sym(\Omega)$ and for some subset $\Gamma \subseteq \Omega$ we have $\Gamma^x = \Gamma$ for each $x \in G$ (such a set is called $G$-invariant) then each element $x \in G$ could be seen as a ...
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29 views

About relation between simple section and section in induced representation on orbits

The following is from Dixon & Mortimer, Permutation Groups, page 74. A group $S$ is called a section of a group $G$ if for some subgroups $H$ and $K$ of $G$ we have $K \unlhd H$ and $H/K \cong ...
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showing $ D_{8} $ is satisfy the permutizer condition

Let $ G $ is a finite group. Permutizer subgroup $ H $ of $ G $ is $ P_{G}(H) = \langle g \in G \ \vert \ \langle g \rangle H = H \langle g \rangle \rangle \ $. If for any proper subgroup $ H $ of $ G ...
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61 views

Prove that the total number of subgroups of a finite cyclic group $G=\langle a \rangle$ such that $o(G)=n$ is the number of divisors of $n$.

Prove that the total number of subgroups of a finite cyclic group $G=\langle a \rangle$ such that $o(G)=n$ is the number of divisors of $n$. If $G=\langle a \rangle $ is a finite cyclic group ...
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1answer
46 views

Sufficient condition to check whether a group is Noetherian

Suppose $G$ is a finitely generated group. What conditions on $G$(or some subgroups of $G$) will force it to be a Noetherian group? Of course, if all subgroups are finitely generated or ACC on ...
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50 views

Prove that a set is a subgroup of $S_G$ (the set of all permutations of a group G)

The question: Suppose that $G$ is a group, and $\forall a\in G$, $f_a:G\to G$ is defined as $f_a(x)=ax, \forall x\in G$. If $S_G$ is the set of all permutations of G, prove that $H=\left\{ f_a:a\in ...
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2answers
17 views

If $|G|=2n$, $|a|=n$, $|t|=2$ and $tat=a^{-1}$ then $t\not \in \langle a\rangle$

Let $G$ be a group of order $2n$ and $a,t\in G$ s.t $|a|=n, |t|=2$ and $tat=a^{-1}$. define $N=\langle a\rangle$. I need to show that $|G:N|=2$ and $G=N\cup tN$. to show that $|G:N|=2$ is trivial ...
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82 views

Let $G$ an commutative group of order $2n$ where $n$ is odd prime. Suppose $G$ has an element of order $n$. Show that $G$ is cyclic.

Let $G$ an commutative group of order $2n$ where $n$ is odd prime. Suppose $G$ has an element of order $n$. Show that $G$ is cyclic. I know that every group of even order has an element of order ...
3
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227 views

Using Lagrange's theorem, prove that a non-abelian group of order $10$ must have a subgroup of order $5$.

Using Lagrange's theorem, prove that a non-abelian group of order $10$ must have a subgroup of order $5$. Attempt: Let $G$ be a group of order $10$. By Lagrange's theorem, if there exist a ...
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51 views

Prove that group $G=\{n_1r_1+n_2r_2 | n_1,n_2\in\mathbb{Z}\}$ is cyclic

$r_1$ and $r_2$ are rational numbers. Prove that the group $G=\{n_1r_1+n_2r_2 | n_1,n_2\in\mathbb{Z}\}$ under addition is cyclic. Generalize to the case where you have $r_1, r_2,....,r_k$ rationals. ...