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 are the Lie subgroups of $\text{U}(3)$?

Does anyone know what the Lie subgroups of $\text{U}(3)$ are?
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64 views

An example where $Z(Z_G(A))$ is not a subset of right Engel elements in a finite $p$-group

Find a counter example to the following statement: Let $G$ be a finite $p$-group such that $G/Z(G)$ has exponent $p$. Let $A$ be a normal abelian maximal subgroup of $G$, and $Z_G(A)$ be the ...
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3answers
387 views

Non trivial group homomorphism from $G$ to $H$

Actual Question is to check if there is : Non trivial group homomorphism $\eta : S_3 \rightarrow \mathbb{Z}/3\mathbb{Z}$. What I have tried so far is : I take $(1 2)\in S_3$, this has order $2$. ...
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2answers
150 views

What is the group of endomorphisms of $\mathbb{Q}/\mathbb{Z}$

As the question says, I'm trying to work out what $End_{\mathbb{Z}}(\mathbb{Q}/\mathbb{Z})$ is. These are just group homomorphisms. But so far all I can see is that its probably enough to see where ...
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5answers
129 views

Contemporary Abstract Algebra (External Direct Products):

My question is from the book (Contemporary Abstract Algebra) in chapter 8 exercise 8: Prove that $S_4$ is not isomorphic to $D_4 \times Z_3$. If anyone if may can help me with this problem I ...
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1answer
148 views

How to prove something about the order of element at a group [duplicate]

$G$ is a cyclic group. the order of $G$ is $n$ ($|G|=n$). $m\mid n$, I have to prove that there is $b\in G$ that $ord(b)=m$. Know, here is one part of the proof: if $m\mid n$ that means that $n=mk$. ...
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1answer
76 views

How to prove or disprove finiteness?

How to prove or disprove that statement: a group is finite if the set of all its subgroups is finite?
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48 views

If $\psi:G \to H$ is a surjective homomorphism, then $|\{g \in G: \psi(g)=h_1\}| = |\{g \in G: \psi(g)=h_2\}|, \forall h_1,h_2 \in H.$

If $\psi:G \to H$ is a surjective homomorphism, then $|\{g \in G: \psi(g)=h_1\}| = |\{g \in G: \psi(g)=h_2\}|, \forall h_1,h_2 \in H.$ Could anyone advise on the proof? If $\psi$ is injective, then ...
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1answer
82 views

Let G be an abelian group and fix a prime p. Prove that G/P has no element of order p.

Let $G$ be an abelian group and fix a prime $p$. Let $P=\{g\in G|\operatorname{ord}(g) \text{ is a power of }p \}$. Let's assume that $P$ is a subgroup of $G$. Prove that $G/P$ has no element of ...
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1answer
41 views

Is there a metabelian group satisfying the following conditions

Is there a group $G$ satisfying the following conditions : (a) $G$ is metabelian of class $p$, whose commutator subgroup has exponent $p$; (b) $G$ has no abelian subgroup of index $p$; (c) There is ...
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1answer
148 views

If a subgroup of a symmetric group has an odd permutation, then it has a subgroup of index 2.

I want to show that for $n\geq 2$ and $H\leq S_n$ if $H$ contains an odd permutation then it necessarily has a subgroup of index 2. I am not sure how to start, if it has an odd permutation then it ...
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3answers
1k views

About stabilizer in group action

Let $X$ be a finite set and $x$ is an element of $X$. Let $G_x$, the stabilizer subgroup, be the subset of $S_X$ consisting of permutations that fix $x$. The question is Is stabilizer always a ...
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2answers
789 views

Why is the kernel of an isomorphism always equal to the identity?

I'm just learning about isomorphisms. Suppose $f:G\rightarrow G'$ is an isomorphism from the groups $G$ to $G'$. Why then is the kernal of $G$ equal to $\{e_G\}$? According to my source, we have ...
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295 views

$G$ is finite, $A \leq G$ and all double cosets $AxA$ have the same cardinality, show that $A \triangleleft G$

If $G$ is a finite group and $A$ is a subgroup of $G$ such that all double cosets $AxA$ have the same number of elements, show that $gAg^{-1}=A$ for all $g \in G$. Here is my attempt, I guess it's ...
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1answer
164 views

Why are projective modules cohomologically trivial?

Let $G$ be a finite group, $H\subset G$ a subgroup, $k$ a commutative ring, $M$ a $kG$-module, $n\in\mathbb{Z}$, and $\hat{H}\,^n(H,M)$ the $n$th Tate cohomology group as defined in this question, ...
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1answer
52 views

Properties of equivalence relations

Let $\sim_1$ and $\sim_2$ be distinct equivalence relations on $A$. Define $\sim_3$ by $a\sim_1 b$ and $a\sim_2 b$. Let $[x]_i$ denote the equivalence class of $x$ for $\sim_i$ ($i=1,2,3$). Prove ...
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1answer
109 views

Finite generation of Tate cohomology groups

Let $G$ be a finite group, and let $F$ be a complete resolution for $G$. In other words, $F$ is an acyclic chain complex of projective $\mathbb{Z}G$-modules together with a map ...
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2answers
134 views

The notation $\mathbb{Z}[\alpha]$

Let $\alpha$ be a real number. I'm studying group theory from the notes of my brother (I'm 16) and I often jump into the notation $\mathbb{Z}[\alpha]$, which, however, is defined nowhere through the ...
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1answer
84 views

How to show that an automorphism of $S_n$ is inner?

Given an automorphism $\phi:S_n\rightarrow S_n$ such that it maps all the transpositions on the transpositions, how do I show that this map is given by a conjugation with an element $s\in S_n$? ...
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1answer
113 views

Groups with 3 elements

I need your help about groups.I know that it is possible to express group of 2 elements. but İs it possible to express group of 3 elements??
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2answers
115 views

$\mathbb{Z}\times \mathbb{Z}_{2}$ is a cyclic group?

I think that $\mathbb{Z}\times \mathbb{Z}_{2}$ isn't a cyclic group becuase we don't have any $(a,b)\in \mathbb{Z}\times \mathbb{Z}_{2}$ that can create the group $\mathbb{Z}\times \mathbb{Z}_{2}$. ...
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108 views

Subgroups of given index of a finitely generated group

Let $G$ a finitely generated group. Prove that for a given natural $n$, there exists finitely many subgroups of index $n$ in $G$. I tried to keep on this approach : Finitely generated group has ...
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3answers
113 views

Cardinalities of Sylow subgroups

I must be missing something very simple here. Problem: Let $|G|=56$. Let H be a Sylow 7-subgroup of G, and suppose H is not a normal subgroup of G. What are the cardinalities of the Sylow ...
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1answer
86 views

When is the action of $G$ on $\text{Syl}_p(G)$ by conjugation is double transitive?

We know that the action of $G$ on $\text{Syl}_p(G)$ by conjugation is transitive. I wonder when this action can be double transitive on $\text{Syl}_p(G)$. Thanks for your help.
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0answers
88 views

How to understand the direct product of group representations (on example)?

The algebra of the Lorentz group $SO(3, 1)$ can be represented as direct product of $SU(2)$ or $SO(3)$ algebras. How to understand this statement?
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1answer
68 views

Abstract Algebra Group theory

Let $G:= \{ e^z \;:\; z\in\mathbb{C}\}$ form a group under multiplication. Question : G is isomorphic with which group?
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3answers
133 views

Product in an non-commutative group

I apologize in advance if my question is stupid. Let $G$ be a group, and $x_1, x_2,\dots,x_n$ a family of elements of $G$. If their product is equal to $e$ (neutral element), is it possible to show ...
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1answer
57 views

Let $p>q$ be primes such that $p \equiv 1 \pmod q$. Then there exists exactly one (up to isomorphism) abelian group of order $pq.$

Let $p>q$ be primes such that $p \equiv 1 \pmod q$. Could anyone advise me on how to show there exists exactly one (up to isomorphism) abelian group of order $pq$? Thank you.
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1answer
396 views

What is the correspondence between structure constants and a Lie group?

Let $T^a$ (with $a = 1,2,\ldots,n$) be a set of generators of a Lie group that satisfy the commutation relations: \begin{equation} [T^a,T^b] = i \sum_{c=1}^n f^{abc} T^c \,, \end{equation} where ...
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1answer
400 views

Taking the automorphism group of a group is not functorial.

Once upon a time I proved that there is no functorial 'association' $$F:\ \mathbf{Grp}\ \longrightarrow\ \mathbf{Grp}:\ G\ \longmapsto\ \operatorname{Aut}(G).$$ A few days ago I casually mentioned ...
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1answer
169 views

Application of Sylow's theorem

Let $p>q$ be primes. $ (1): \exists $ non-abelian group of order $pq$ $\Longleftrightarrow$ $p \equiv 1 (mod \ q)$ $(2):$ Any $2$ non-abelian groups of order $pq$ are isomorphic to each other. ...
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6answers
447 views

If $G$ is a group and $N$ is a nontrivial normal subgroup, can $G/N \cong G$? [duplicate]

I know $G/N$ is isomorphic to a proper subgroup of $G$ in this case, so the gut instinct I had was 'no'. But there are examples of groups that are isomorphic to proper subgroups, such as the integers ...
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1answer
43 views

I need a proof of a theorem about generated group

how to use definition 2.7 to prove theorem 2.8?
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2answers
366 views

find all the members of A4/K

Let K={(1),(12)(34),(13)(24),(14)(23)} show that K is a normal subgroup of the alternating group of degree A4 and find all the members of A4/K and write down the group table for A4/K. i know that K ...
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1answer
46 views

Request for the proof of a theorem

I am sincerely requesting a proof of the following theorem from anyone willing to do so. We are about to learn about Sylow groups and simple groups next, and I want to get further ahead of the game by ...
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1answer
761 views

How to prove that a group of order $72=2^3\cdot 3^2$ is solvable?

Let $G$ be a group of order $$72=2^3\cdot 3^2$$ Without using Burnside's Theorem, how to show that $G$ is solvable? Atempt: If we can show that $G$ has at least one non-trivial normal subgroup $N$, ...
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2answers
192 views

Give an example of a nonabelian group such that G/Z(G) is… [closed]

A) abelian; B) nonabelian; Not sure here.
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1answer
110 views

Help with understanding non-isomorphic groups

This is a question from my course, which I am having problems understanding. For each integer $n>1$, give examples of non-isomorphic groups of order $n^2$.
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1answer
135 views
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1answer
55 views

Group Isomorphism Problem

Let $S=\{(k,x) | k \in \mathbb{F}_5 \setminus \{0\}, x \in \mathbb{F}_5 \} $ be the group with binary operation $(k,x)*(l,y)=(kl,xl+y)$. Let $P$ be a Sylow 5-subgroup of $G=Sym(5)$. I am asked to show ...
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1answer
53 views

Question about $e$ element at $\mathbb{Z}_{n}$

At group $\mathbb{Z}_{n}$, $e=0$? I assume that is true but I just what to know if I'm right. Because for every $a\in \mathbb{Z}_{n}, a^0=a\cdot 0=0$. Thank you!
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2answers
239 views

Isomorphism types of semidirect products $\mathbb Z/n\mathbb Z\rtimes\mathbb Z/2\mathbb Z$

Let $n = p_1 p_2 \cdots p_k$ be the product of pairwise distinct odd primes. Let $X$ be the set of element in $\operatorname{Aut}(\mathbb Z/n\mathbb Z)$ of order $1$ or $2$. For each $\psi\in X$, ...
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Questions about some subgroups…

I want to ask if I'm understand few subgroups correct: $\langle 2\rangle$ of $\mathbb{R}$ is $\left\{2\cdot n\big|n\in \mathbb{Z} \right\}$ $\langle 2\rangle$ of $\mathbb{R}^{*}$ is ...
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1answer
412 views

The intersection of two Sylow p-subgroups has the same order

Let $G$ be a finite group and assume it has more than one Sylow p-subgroup. It is known that order of intersection of two Sylow p-subgroups may change depending on the pairs of Sylow p-subgroups. ...
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3answers
1k views

Classification of groups of order 30 [duplicate]

How do I find all the groups of order 30? That is I need to find all the groups with cardinality 30. I know Sylow theorems.
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1answer
88 views

Let $\phi:G \to (\mathbb{Z}/(15),+)$ be a surjective homomorphism. Then, $G$ has normal subgroups of indices $3$ and $5.$

May I verify if my proof is correct? Thank you. Let $\phi:G \to (\mathbb{Z}/(15),+)$ be a surjective homomorphism. Then, $\exists H_1 \lhd G, H_2 \lhd G$ such that $|G:H_1|=3$ and $|G:H_2|=5.$ ...
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59 views

Product of three supersoluble subgroups

I need a proof of the following statement: Let a finite group $G=AB=AC=BC$ be the product of three supersoluble groups $A$, $B$ and $C$. If the commutator subgroup $G'$ is nilpotent, then $G$ is ...
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1answer
55 views

Group isomorphic to (Z,+)

Please I beg help on this one. Even the first case its hard for me to prove Consider $a\in\Bbb R$ such that $a \ne 0,1,-1$ and let $H=\{a^n \mid n\in\Bbb Z\}$. Show that $(H,\cdot)$ is a group ...
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1answer
249 views

H acting on G by left translation

$H$ is a subgroup of $G$ and acts on $G$ by left translation, describe the orbits. Here is my take, at first I thought well, isn't that just left cosets? But, that seems too easy. By translations, ...
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103 views

Prove or disprove: There exists a set of two elements which generates the group $\mathbb{Z}^3$.

Prove or disprove: There exists a set of two elements which generates the group $\mathbb{Z}^3$. I think the statement is NOT true. If we treat $\mathbb{Z}^3$ as a vector space, then its dimension ...