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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Conjugate diagonal matrices

Let $M$ be the set of matrices that have precisely one entry in each row/column that is nonzero and $D$ be the set of invertible diagonal matrices ( so all entries down the diagonal are nonzero). ...
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Hom$_{Set}(G \times H,R) \cong $Hom$_{Set}(G,R)\otimes $ Hom$_{Set}(H,R)$?

If $G$ and $H$ are finite groups and $R$ is a ring does Hom$_{Set}(G \times H,R) \cong $Hom$_{Set}(G,R)\otimes $ Hom$_{Set}(H,R)$ if Hom sets have the usual R-algebra structure?
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$G$ is direct product of $H_1,\ldots,H_k$

Let $G$ be a direct product of the groups $H_1,\ldots,H_k$, let $K\vartriangleleft G$. Prove that if $Z(G)$ doesn't contain $K$, then there is $i$ such thet $K\cap H_i \neq\{ e \}$ ($e$ is the ...
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confusion about cosets and quotient space

The set of cosets $V/U = \{ v+U: u \in V \}$ with operations $$(v+U) + (w+U) = v+w+U$$ $$a(v+U) = av+U$$ (which is well defined) is a vector space called Quotient Space. I am having a difficult ...
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How can I understand $\Bbb Z\times \Bbb Z/2\Bbb Z$

This may be stupid request, but I would like to have a intuition for the group $\Bbb Z\times \Bbb Z/2\Bbb Z$ in terms of 'real' objects. 'Real' could mean geometric but not necessarily. I perhaps what ...
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A group which is $\mathbb{Z}$-by-finite but not finite-by-$\mathbb{Z}$

I found a lemma which states: If a group $G$ is finite-by-$\mathbb{Z}$, then $G$ is $\mathbb{Z}$-by-finite. I was wondering if the converse is true, i.e. is it true that if a group G is $\...
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Show that $\mathbb{Z}_9$ is not the homomorphic image of $\mathbb{Z}_3 \times \mathbb{Z}_3$.

We solved a similar question in class: $\mathbb{Z}_8$ is not the homomorphic image of $\mathbb{Z}_{15}$ because if $f \colon \mathbb{Z}_{15} \to \mathbb{Z_8}$ is a homomorphism, then $|Im(f)|$ ...
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41 views

Question about normal subgroups of a group

I have a group $G$ and two normal subgroups $N_1, N_2 \unlhd G$ with $N_1 \leq N_2$. Is it true that $G/N_2 \leq G/N_1$?
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How is this a counter example to “ $A$ and $B$ are isomorphic but $G/A \ncong G/B”$?

If $G = \mathbb{Z}_2+ \mathbb{Z}_4$, and $A=\langle(1, 0)\rangle$ and $B=\langle(0, 2) \rangle$, how does it follow $G/A \ncong G/B$. How do we know $A$ and $B$ are normal subgroups of $G$?
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Application of First Isomorphism theorem - Help with finding Kernel

Are the groups $\Bbb{R}/\Bbb{Z}$ and $S^1$ $isomorphic$? Here $\Bbb{R}$ is the additive group of real numbers, and $S^1 = \{z\in\Bbb{C}\mid |z| = 1\}$ under complex multiplication. I am not able to ...
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31 views

Show homomorphism of semidirect product

Given: $G=NH$ and $N\cap H=\{e\}$ Show the map $\alpha:H\rightarrow\text{Aut}(N)$, given by $\alpha(h)=\alpha_h$, where $\alpha_h(n)=hnh^{-1}$, is a homomorphism Do I have to show $\alpha(h_1h_2)=\...
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0answers
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Show the map from the product of disjoint subgroups of G to G is an isomorphism [duplicate]

Question: Show that if $G$ is a group with two normal subgroups $H$ and $K$ such that $G=HK$ and $H\cap K=\{e\}$, then the map $(h,k)\mapsto hk$ is an isomorphism of groups from $H\times K$ to $G$ ...
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109 views

Find a homomorphism of general linear group whose kernel consists of matrices with positive determinant

Let $G=GL_n(\mathbb{R})$, the group of invertible $n\times n$ real matrices, Let $N$ be the subset of $G$ given by $\{A\in G:\det(A)>0\}$ Let $H=\{1,-1\}$ considered as a group under ...
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Frattini subgroup of a $p$- group of order $p^4$

Let $p$ be an odd prime and $G$ be a finite non-abelian $p$-group of order $p^4$ with the following presentation: $$\langle a, b, c, d\mid a^p=b^p=c^p=d^p=1, c^d=cb, b^d=ba, [a,d]=[b,c]=[a,c]=[a,b]=1\...
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147 views

Automorphism group of a semi-direct product

I'm trying to construct the semi-direct product $(\mathbb{Z}_7 \rtimes \mathbb{Z}_3) \rtimes \mathbb{Z}_2$. Constructing the first factor in parentheses is not difficult. But when it comes to ...
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Cardinality of set of groups

After analyzing this question I started wondering. Thoughts. Everyone can give a simple example of a countable group $G, |G| = \aleph_0$ which has uncountable number $2^{\aleph_0}$ subgroups, for ...
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101 views

A group of order $12$ either has a normal $ 3$-Sylow subgroup or is isomorphic to $A_4$

Let $G$ be a group of order $12.$ Prove that either $G$ has a normal $ 3$-Sylow subgroup or $ G$ is isomorphic to $A_4$. I know that $|G|=12=2^23$ and that either $n_3=1$ and there is a $3$-sylow ...
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Question on unipotent radical

If $G$ is a linear algebraic group then its unipotent radical $R_u(G)$ is normal in $G$. Is it also contained (and normal) in $G^0$?
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Let $G$ be a group of order 315 with a normal 3-Sylow subgroup. Prove $G$ is abelian.

I know this it a prevalent question, I really do. It's just that every proof requires using Automorphisms groups about which we were barely taught. I can't start learning everything about ...
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Harmonic Analysis on the Affine Group

In my previous question, I asked about harmonic analysis on the group $\operatorname{SL}(3, \mathbb{R})$. The representation theory of this group appears to be quite complicated, so I am now looking ...
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The group $E(\mathbb{F}_p)$ has exactly $p+1$ elements

Let $E/\mathbb{F}_p$ the elliptic curve $y^2=x^3+Ax$. We suppose that $p \geq 7$ and $p \equiv 3 \pmod {4}$. I want to show that the group $E(\mathbb{F}_p)$ has exactly $p+1$ elements. I was ...
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Is the quotient of connected linear algebraic groups connected?

Let $K$ be a field (of characteristic zero if it makes things nicer) and $G/K$ a connected linear algebraic group (i.e. smooth affine). Let $N \leq G$ be a connected normal linear $K$-subgroup of $G$. ...
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How to count the elements of order 196 in a given abelian group

The abelian group $A_2$ is defined as $C_4 \times C_4 \times C_{49} \times C_7$ and I need to find the number of elements of order $196$ in $A_2$. I understand that I must take an element of order $...
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150 views

Rank nullity theorem

Let $$0\rightarrow G_{1}\xrightarrow{f_{1}} G_{2}\xrightarrow{f_{2}} G_{3}\rightarrow 0$$ be a short exact sequence of finitely generated abelian groups. We call $\overline{G_{i}}$ the quotient of $G_{...
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75 views

Element commuting with normal subgroup of $p$-group

Let $p$ be a prime. Suppose $N\triangleleft G$ where $|G| = p^n$ ($n>2$) and $|N| = p$. Prove there exists $g\not\in N$ such that $gn = ng$ for all $n\in N$. I am supposed to prove this without ...
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Given a non-zero element of $S_n$ can we find another so that together they generate $S_n$?

Let $\sigma$ be a non-identity permutation of $S_n$, can we find an element $\rho$ in $S_n$ so that $\{\sigma\}\cup\{\rho\}$ are generators of $S_n$? It is known that $S_n$ is generated by $(12)$ and $...
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38 views

For a special group G described below, the order of G can only be $p_1p_2…p_k$ where $p_i$ s are prime.

I am just learning generators and relations basics. A clarification Let $G$ be a finite group in which every element(non-zero) is a possible candidate to be in the minimal generating set $S$ of the ...
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When does $S_n$ have exactly one subgroup of index $2$?

When $n\geq 5$ we can assume there is another one besides $A_n$, they are both normal, so the intersection is normal in $S_n$ and also in $A_n$, contradicting that $A_n$ is simple. For $n=1$ it's ...
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Normalizer of a subgroup generated by a cycle.

Let $H$ be the cyclic subgroup of $S_4$ generated by the cycle $(1234)$. What is the order of the normalizer $N$ of $H$ in $S_4$? Give generators for $N$. How do I go about solving a ...
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Group Theory Isomorphism

Show that if $G$ is a group with two normal subgroups $H$ and $K$ such that $G=HK$ and $H\cap K=\{e\}$ then the map $(h,k)\rightarrow hk$ is an isomorphism of groups from $H\times K$ to $G$. I am not ...
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1answer
90 views

Compact objects and locally finitely presentable categories (the Category of Groups)

I am trying to understand the concept of locally finitely presentable categories. I have discovered the concept of compact object here. I have discovered that for groups, the finitely presented ...
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Find the number of subgroups of $\mathbb{Z}/(5)\times \mathbb{Z}/(5)$

Find the number of subgroups of $\mathbb{Z}/(5)\times \mathbb{Z}/(5)$ including trivial subgroups. My Work: If we consider $\mathbb{Z}/(5)$ then the only subgroups are trivial subgroups. But how ...
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72 views

Exact sequence of finitely generated abelian groups

Let $$0\rightarrow G_{1}\xrightarrow{f_{1}} G_{2}\xrightarrow{f_{2}} G_{3}\rightarrow 0$$ is a short exact sequence of finitely generated abelian groups. We call $\overline{G_{i}}$ the quotient of $G_{...
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Are the primitive groups linearly primitive?

A transitive permutation group $G \subset S_n$ is primitive if $G_1 \subset G$ is a maximal subgroup. A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation. ...
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Non exact sequence of quotients by torsion subgroups

$$0\rightarrow G_{1}\rightarrow G_{2}\rightarrow G_{3}\rightarrow 0$$ is a short exact sequence of finitely generated abelian groups. We call $\bar{G_{i}}$ the quotient of $G_{i}$ by its torsion ...
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Find all the $p$-Sylow subgroups of $D_6$.

$|D_6|=12=2^23.$ I started with $3$. I know that the number of $3$-Sylow subgroups, denoted $n_3$, is: $1,4,7...$ and I also know that $n_3|2^2$. e.g, $n_3=1, 4$. How can I show that it can't be $4$? ...
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40 views

let $G$ to be group such that $O(G)=p^2$ where $p$ is prime,prove that $G$ is cyclic or $G$ is Direct product of two cyclic subgrops of order $n$. [duplicate]

the only hint that i got is Sylow's first theorem, which implies that if $p^n$ is any prime power dividing $O(G)$, then $G$ has a subgroup of order $p^n$. in our case $p$ devides $p^2$, then we can ...
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How to implement a group action in Sage ( for educational purposes )?

Let S= {"A","B","C","D"} and S4= SymmetricGroup(4). I want to create a table of the action S4 x S -> S which standardly permutes the letters in the set. The table should look like: ...
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Subgroups of Semidirect Product of the elementary abelian group of order 8 by $S_3$

What are the subgroups of the semidirect product of the elementary abelian group of order 8 by $S_3$? This is the group $(\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2)\rtimes S_3$ of order 48; ...
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If a group has one p-sylow subgroups, then this subgroup must be normal.

I've learned that this is true. Why, basically? I'd appreciate you help.
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Canonical form by J. A Green

Let $f(x)=x^d-a_{d-1}x^{d-1}\cdots-a_0$ be a irreducible polynomial over finite field $\mathbb{F}_q$, $C(f)$ be its company matrix. Let $A$ be a matrix with minimum polynomial $m(x)$. For the sake of ...
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Which inclusions of finite groups are relatively linearly primitive?

This post is a sequel of: Which finite groups have faithful complex irreducible representations? A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation. ...
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A p-Sylow subgroup of a subgroup is a p-sylow subgroup of the group

Is that true? If it is, then it is unprovable. I guess it isn't. The question is: how can I show it unequivocally? The definition my teacher gave us is: $P$ is a $p$-Sylow subgroup of $G$ if $|P|=...
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subgroup is saturated iff it is a direct summand

Let $A$ be a free abelian group of finite rank. Call a subgroup $B \le A$ saturated if for all $a \in A$ and positive integers $n$ such that $na \in B$, the element $a$ belongs to $B$. Call a subgroup ...
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291 views

A group of order $pqr$ (primes $p > q > r$) has a subgroup of order $qr$ [duplicate]

I've done most of the following problem, but I can't seem to get part (d). Let $G$ be a group of order $pqr$ for primes $p > q > r$. By a counting argument one can see that there is ...
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50 views

$G$ is characteristically simple $\iff$ there is simple $T$ such that $G \cong T\times T \times \cdots \times T$

Let $G$ be a characteristically simple finite group, i.e. it has no nontrivial characteristic subgroups. Prove there is some simple group $T$ such that $G \cong T \times T \times \cdots \times T$. ...
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85 views

How many normal subgroups?

$(a)$ Let $T_1$, $T_2$, $\cdots$, $T_k$ be non-abelian finite simple groups. How many normal subgroups does the direct product $T_1 \times T_2 \times \cdots \times T_k$ have? $(b)$ Let $G$ denote ...
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256 views

The finite groups with an irreducible faithful complex representation

All the groups below are supposed finite, and their representations, complex. An abelian group admits an irreducible faithful representation iff it is cyclic. A group has all its non-trivial ...
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115 views

Automorphism group of a tournament is solvable

$(a)$ Let $X$ be a tournament, i.e. $X$ is a directed complete graph. Denote $V(X)$ the vertex set of $X$. An automorphism of $X$ is a bijection $V(X) \to V(X)$ preserving orientation. Prove that the ...
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What is the most mundane & intuitive meaning of the radical of a finite group?

I am asking this question because I am trying to solve this problem from a class note: Assume that $R(G)$ is simple and not commutative, show that $G$ is a subgroup of $Aut(R(G)).$ The note's ...