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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What can we say about the order of the group which is generated by $\langle u,v\rangle$

If a group $G$ is generated by $\langle u,v\rangle$ and order of $u$ is less than or equal to $4$ and order of $v$ is less than or equal to $3$. Then what can we say about the order of the group? It ...
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Product of all elements in finite abelian group equals its own inverse

Let $G = \{e, a_1, a_2, \dots, a_n\}$ be a finite abelian group and let $S$ be the set of all the elements of $G$ which are not equal to their own inverse. The set $S$ can be divided up into pairs so ...
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Finding the inverse of an element of $S_n$ and it's order [duplicate]

I have two questions, 1) What are the ways to find the inverse of an element of $S_n$? 2) What are the ways to find the order of an element of $S_n$?
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Prove that $\langle S \rangle = G$

Let $S\subset G$ be a finite group such that $\#2S\gt\#G$ Prove that $\langle S \rangle = G$ My idea was to start by taking $a \in G$ and proving that $ax^{-1}$ is also in $G$ if $x$ is in $S$, but ...
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55 views

Isomorphic relation between dihedral groups

Theorem Let $G,H$ be abelian groups such that $Dih(G)\cong Dih(H)$ If $G$ is finitely generated, then $G\cong H$. I'm curious whether "finitely generated" hypothesis can be removed. If it'...
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Fundamental representation of $O(3)$

I want to check if the fundamental representation of $O(3)$ is irreducible on $\mathbb{R}^3$ and $\mathbb{C}^3$. I want to use isomorphism properties. I know this relation exists $$ \mathfrak{o}(3)\...
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General Classification of finite simple ternary groups?

Define a ternary group as an algebraic set endowed with a 3-ary operation f: that maps 3 elements onto another in the set. Furthermore for any three elements a,b,c there exists a unique 4th element ...
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Isomorphic but not equivalent actions of a group G

This is in some sense a continuation of this problem. Given a group $G$ I would like to exhibit two actions of $G$ on a set $[n] =\{1,\ldots,n\}$ such that the two actions are isomorphic yet not ...
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145 views

Finite groups with nontrivial outer automorphisms

Let $G$ be a finite group whose center $Z(G)$ is trivial. Suppose that the group $\text{Out}(G)$ of outer automorphisms is nontrivial. Question: Does there always exist an $f \in \text{Aut}(G)$ ...
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When is a group the product of its subgroup and their factor group?

If G is a group and H a normal subgroup of G, under what conditions is it true that $$H \times (G/H) =G ?$$ To be more specific: Characterise groups G which allow proper non-trivial normal ...
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123 views

Commutator subgroup of matrices of order two!

Let $G$ be the group of all invertible matrices on the real field of order two under multiplication. And define $N=\{X\in G:det(X)=1\}$, prove that $N=G'$. That's easy to prove that $G'\subset N$ ...
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33 views

How can I find the subgroups of $D_5$

So I've found all the cyclic subgroups: $\langle e\rangle$, $\langle r\rangle$, $\langle sr^n\rangle$, and $D_5$ itself (which are 8 subgroups), but how do I know if these are all? How can I find the ...
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51 views

Determine if $\mathbb{Z}/12\mathbb{Z}\setminus\{0\}$ is a group under usual product.

I'm doing an exercise about determining if some sets with binary operations have group structures. I'm struggling with this one: The set $\mathbb{Z}/12\mathbb{Z} \setminus \{0\}$, with the usual ...
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35 views

All distinct groups of $C_{13} \rtimes_{h} C_4 $

Describe all homomorphisms $h: C_4 \to Aut(C_{13})$, for each h describe $C_{13} \rtimes_h C_4 $ in terms of generators and relations. How many distinct isomorphism types of groups of the form $C_{13} ...
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59 views

Order of Aut$(D_4)$

How can I prove that order of Aut$(D_4)$ is 8. Let we show $D_4$ as $\{e,\sigma,\sigma^2,\sigma^3,\tau,\tau\sigma,\tau\sigma^2,\tau\sigma^3\}$ and $\quad\sigma^4=e=\tau^2,\quad\sigma\tau=\tau\sigma^3$....
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18 views

Presentation of $Dih_n$

Let $\varphi:Z_2\rightarrow Aut(Z_n)$ be a homomorphism such that $\varphi(\overline{1})$ is the automorphism by inversion. Set $Dih_n\triangleq Z_n\rtimes_{\varphi} Z_2$. How do I prove that $<x,...
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Reducibility of Cyclic groups

Let $G$ be the cyclic group $C_{4}$ and consider the 2-dimensional representations of G. Why does extending scalars to the complex numbers let this representation become reducible? I understand how it ...
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443 views

Let $A,B$ be subgroups of a group $G$. Prove $AB$ is a subgroup of $G$ if and only if $AB=BA$

Let $A,B$ be subgroups of a group $G$. Prove $AB$ is a subgroup of $G$ if and only if $AB=BA$ Could someone explain/define the multiplication here for me so that I may attempt this problem. Thank you ...
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Definition of minimal presentation of a group

I'm working on a problem on the braid monodromy of complex lines arrangements in $\mathbb{C}^{2}.$ I have the following question. It's just a simple definition. However, I didn't find anywhere. Let $...
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Mellin transform defined for function on group $(\mathbb{R}^{+}, \times)$

In the article on the $\Gamma$ function in the Princeton Companion to Mathematics, the author states The Mellin transform is a type of Fourier transform, but it is defined for functions on the ...
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Let $H$ be a subgroup of a group $G$ and $a,b \in G$. Prove that the following

Let $H$ be a subgroup of a group $G$ and $a,b \in G$. Prove that the following statements are equivalent. (1) $a^{-1}b \in H$ (2) $b^{-1}a \in H$ (3) $aH=bH$ So, I started with (3). $aH=bH \to a^{...
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Prove: $aH = bH \iff Ha^{-1} = Hb^{-1}$

I have to prove this exercise for my math-study: Let $G$ be a group and $H \subset G$ a subgroup. Prove that for every $a,b \in G$ holds: $$aH = bH \iff Ha^{-1} = Hb^{-1}$$ I tried this, but I'm ...
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110 views

Let $S$ be a completely regular semigroup, expressed as a semilattice $Y$ of completely simple semigroups $S_{\alpha}$ ($\alpha \in{Y})$

Show that, if $L$ is a left ideal of $S_{\alpha}$, then $$L \cup [\cup \{S_{\beta} : \beta < \alpha\}]$$ is a left ideal of $S$. Suppose now that $S= \mathcal{S}(Y;G_{\alpha};{\phi}_{\alpha,\...
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49 views

$G=HK$ then the index of a subgroup is determined by $H$ and $K$

Let $G=HK$ s.t. $H\cap K=1$ and let $R$ be a any subgroup of $G$. I wonder necessary and suffucient condition for the equality, $$|G:R|=|H:H\cap R||K:K\cap R|$$ Note that if $H$ and $K$ are normal ...
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63 views

Left orderable Group has infinite order

An $order$ on a set $S$ is an (anti-symmetric) relation $<$ on $G$ so that for each $a,b\in G$ exactly one of the following is true: $a<b, b<a$ or $a=b$. A group $G$ is called left orderable ...
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Independent components of a group cocycle

Fix a finite group $G$. An $n$-cochain of $G$ with coefficients in a $G$-module $M$ is a function$$b:G^n\rightarrow M$$To determine $b$, one must specify the values $b(g_1,\ldots,g_n)\in M$ for all $(...
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Do groups, rings and fields have practical applications in CS? If so, what are some?

This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life (academic and professional). Almost everything I studied I've observed be ...
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ring theory questions… what is a subring?

So I've got coursework to do, and having not been to some (most) lectures, I'm at that time where it's time to learn everything I need to know... Any help is much appreicated, thanks! I've been given ...
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$\left\langle\begin{pmatrix}0&-1\\1&0\end{pmatrix},\begin{pmatrix}1&1\\0&1\end{pmatrix}\right\rangle$ is the free product of two cyclic groups.

Let $G$ be a group generated by two matrices $ S=\left( \begin {array}{cc} 0&{-1}\\1&0\end{array}\right),T=\left( \begin{array}{cc}1&1\\0&1\end{array}\right) $ in $SL_2(\mathbb Z)/ \{ \...
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Finite locally groups

Let be $V\leq \operatorname{Aut}\left( G\right),\ N\vartriangleleft G$ and $V$-invariant. Consider the semidirect product of $V$ with $G$. Let $C_{G}(V)=\{g\in G:g^{v}=g\}$ and $[G,V]=\langle g^{-1}...
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A semigroup with identity having exactly one idempotent is a group

Let $G$ be a finite semi-group with identity such that it has only one idempotent.Is $G$ a group? It only remains to show that for any $a\in G$ $\exists b\in G$ such that $ab=ba=e$ where $e$ is the ...
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Proving an isomorphism regarding $\mathbb{Z}$.

I am to show that for all $m,n \in \mathbb{Z}$, the following isomorphism: $\dfrac{(m,n)\mathbb{Z}}{m\mathbb{Z}} \cong \dfrac{n\mathbb{Z}}{[m,n]\mathbb{Z}}$ where $(m,n) = gcd(m,n)$ and $[m,n] = ...
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How is the entire $SO(2)$ group the standard rotation matrix?

In a book I am using, the following is presented, $$\mathcal{R}(\phi) = \begin{pmatrix} \cos (\phi ) &\sin (\phi ) \\ -\sin (\phi ) &\cos (\phi )\end{pmatrix}$$ The group's name is $\...
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If a and b are in G and ab=ba, then we say a and b commute. Assuming a and b commute, prove:

If $a$ and $b$ are in a group $G$ and $ab=ba$, show that $xax^{-1}$ commutes with $xbx^{-1}$ for any $x \in G$. So I wrote: WWTS: $\bf{xax^{-1} \times xbx^{-1}=xbx^{-1}\times xax^{-1} }$ Now, the ...
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217 views

How to show that this Cayley Table does not form a group

Given the following Cayley Table (where e is the identity element): How would I go about proving that the table does not form a group? I have checked closure, identity, inverses, and all 27 ...
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Number of cyclic subgroups order $p^2$ in $\mathbb{Z}_p \times \mathbb{Z}_p \times \mathbb{Z}_{p^2}$

Let $$G={ {<a>}_{p} \times {<b>}_{p} \times {<c>}_{p^2}} \cong \mathbb{Z}_p \times \mathbb{Z}_p \times \mathbb{Z}_{p^2} \text{, $p$ is prime}$$ There are $p^3-1$ elements with order ...
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If $f:G \to H$ is a homomorphism with kernel $N$ and $K$ is a subgroup of $G$ then prove that $f^{-1}(f(K))=KN$

If $f:G \to H$ is a homomorphism with kernel $N$ and $K$ is a subgroup of $G$ then prove that $f^{-1}(f(K))=KN$ Ok, so what I know from this: $G$ and $H$ are groups that must preserve operation. It ...
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Can this lattice be realized as an intermediate subgroups lattice?

Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$. Let the lattice $\mathcal{L}$ as follows: ...
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is the group $(\mathbb{Z}_n \times \{-1,1\}, *) \simeq D_n$

is the group $(\mathbb{Z}_n \times \{-1,1\}, *) \simeq D_n$ $*: (a,b) * (c,d) = (a+_n(c \cdot_n b),bd)$ Showing $(\mathbb{Z}_n \times \{-1,1\}, *)$ was a group was a previous question but I have ...
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Subgroups of group of characters of a finite abelian group

Let G be a finite abelian group, H a subgroup of the group of characters of G. Is it true that H is the group of characters of some quotient group of G? Thanks for any help.
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Representation theory

I am trying to study representation theory from the book Algebra by Artin. I came across the following problem which seemed interesting: Prove that the linear operator $T=\sum_{g\in C} \rho_{g}$ is $...
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A normal intermediate subgroup in $B_3$ lattice?

Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of intermediate subgroups between $H$ and $G$. An intermediate subgroup $H \subset K \subset G$ is a ...
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A normal intermediate subgroup in $B_3$ lattice with an additional index condition?

This post is a sequel of: A normal intermediate subgroup in $B_3$ lattice? Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of intermediate subgroups ...
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If $o(a),o(b)\gt 1$ and $o(a)$ and $o(b)$ are co-prime then $o(a)o(b)$ divides $|G|$

I'm trying to prove the following statement: Let $G$ be a finite group and $a,b\in G$ such that $o(a),o(b)\gt 1$. Prove that if $o(a)$ and $o(b)$ are co-prime then $o(a)o(b)$ divides $|G|$. Well, I'...
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$p^nm$ group, element of order $m$

Let $p$ be an abelian group of order $p^nm$, $p$ prime, and $p$ does not divide $m$. Is it true that the group must contain an element of order $m$, or a multiple of $m$? If yes, how to prove it? If ...
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Prove that the group isomorphism $\mathbb{Z}^m \cong \mathbb{Z}^n$ implies that $m = n$

I tried using a contrapositive, ($m \neq n$ implies $\mathbb{Z}^m \ncong \mathbb{Z}^n$), and I think the problem is that there won't be a homomorphism, but did not get anywhere. Is there a better ...
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119 views

Cyclic properties of multiplicative group G of all the complex $2^n$ roots of unity

Consider the multiplicative group G of all the complex $2^n$ roots of unity, $n=0,1,2,\ldots$ I am asked to verify whether $G$ is a cyclic group and whether it has a finite set of generators. The ...
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81 views

what is the minimum number of conjugacy classes for a group of order $n!$?

i became curious about how one might measure the extent to which a given finite group departs from perfect commutativity. a rough-and-ready index of the degree to which a group $G$ is commutative may ...
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52 views

A necessary and sufficient condition for $\langle (g,h) \rangle = \langle g \rangle \oplus \langle h \rangle$

Let $G$ abd $H$ be finite groups and let $(g,h) \in G \oplus H.$ Then what is a necessary and sufficient condition for $\langle (g,h) \rangle = \langle g \rangle \oplus \langle h \rangle$ Attempt of ...
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$\phi:M \rightarrow N$ is surjective, then $\hat{\phi}:M/Tor(M) \rightarrow N/Tor(N)$ is surjective

Hello everyone, I already have done that $\phi(Tor(M)) \subset Tor(N)$. I'm stuck on the second part, so this is my attempt. Since $\phi$ is an isomorphism, then since $Tor(M)$ and $Tor(N)$ are ...