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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63 views

On direct sum and direct product of groups

I understood the definitions of direct sums and direct products of groups (rings, modules, etc). The definitions of them can be given in terms of universal properties - certain commutative diagram in ...
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1answer
57 views

Determine abelian groups with 48 elements

I am just doing some revision for my linear algebra exam, and I came across this problem: Determine all abelian groups (up to isomorphism) with exactly 48 elements. I am not sure I have ever ...
3
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1answer
32 views

For prime $p$, let $G$ be a group such that every non-identity element of $G$ has order $p$. Show that if $|G|$ is finite, then $|G| = p^n$.

I've been self teaching myself some topics in preparation for university and thought I'd have a go at some past paper questions from their website. As such I do not have much experience with these ...
4
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2answers
105 views

Why are Lie groups automatically analytic manifolds?

In the book by Kolar, Michor, and Slovak, it is shown that multiplication $\mu:G\times G\to G$ is analytic in some neighborhood of $e$. Specifically, they show that in the chart given by $\exp^{-1}$, ...
0
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2answers
36 views

What is $pA$ when $p$ is a prime number and $A$ an abelian group?

Let $A$ be a finite abelian $p$-group. I want to prove that $pA$ is also an abelian finite $p$-group, of order strictly less than the order of $A$. The problem is that I don't even know what does the ...
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49 views

Proving Sylow theorems without using group actions

Most of the proofs of Sylow theorems involves groups actions in some way as below: Sylow theorems - wiki There is a thread for them here, too: Proofs of Sylow theorems. However, I would like to see ...
1
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0answers
17 views

real irreducible representation of SU(2) group

Consider a real irreducible representation of SU(2) group in d-dimensional space-time. How many components do the spinors (eigenvectors) have ? For instance, a real irreducible spinor in 10-dim has 16 ...
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38 views

Subgroups of $(\mathbb{Z}\times \mathbb{Z}) * \mathbb{Z}/2\mathbb{Z}$

I can't figure out who the group $(\mathbb{Z}\times \mathbb{Z}) * \mathbb{Z}/2\mathbb{Z}$ is. (Where $*$ means the free product). In particular I would like to study all its subgroups. Is it true ...
2
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1answer
29 views

Problem with the proof of abelian finite groups decomposition

I can't understand the proof which says that every finite abelian $p$-group can be written as a direct sum of cyclic $p$-groups. I'm using Lang's book of Algebra. My problem is about the following ...
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1answer
40 views

Surjective group homomorphism $f:A_4\times \mathbb{Z}_2 \to S_3$ [closed]

Is there a surjective group homomorphism $f:A_4\times \mathbb{Z}_2 \to S_3$? I need to find one if the answer is "yes" or explain why the answer is "no". With $\mathbb{Z}_n$ I mean the quotient ...
0
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0answers
23 views

Explicitly calculating the group of global isometries of a convex regular polygon (dihedral group)

Instead of an intuitive geometric description of the dihedral groups $D_{2n}$, that one can find in virtually every good book on group theory, I want to calculate the global isometries of a convex ...
0
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2answers
27 views

Cyclic subgroup proof question

Let $a$ be an element of order $n$ in a group and let $k$ be a positive integer. Then $\langle a^k \rangle = \langle a^{\gcd(n,k)} \rangle \text{ and } |a^k| = n / \gcd(n,k).$ The proof starts by ...
3
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2answers
44 views

Prove /Disprove: Existence of Symmetric Subgroup of Order $n^2-1$ of $S_n$.

$G$ is a symmetric subgroup of symmetric group $S_n$ acting on $n$ objects where order of $G$ is $n^2-1$. Problem: Prove (or disporve), Such $G$ exists for $S_n$ for finite $n$. For $n=5,11,71$, ...
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3answers
39 views

If $f:B\to C$ is a homomorphism and $g:A\to B$ is a monomorphism then is $\ker f\cong \ker f\circ g$?

Let $A,B,C$ be groups. Let $f:B\to C$ be a group homomorphism and let $g:A\to B$ be an injective group homomorphism. Is it true that $\ker f\cong \ker f\circ g$? My attempt : Define $\phi:\ker ...
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3answers
43 views

Determining if a factor group is isomorphic to $\mathbb{Z}_4$ or $\mathbb{Z}_2\times \mathbb{Z}_2$

Let $G = \Bbb Z_{4} \times \Bbb Z_{4} $, and $H = \{(0,0), (2,0), (0,2), (2,2)\}$. Is $G/H$ isomorphic to $ \Bbb Z_{4} $ or $\Bbb Z_{2} \times \Bbb Z_{2} $? I know $G/H$ has order $4$. Also, every ...
1
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0answers
8 views

Pronormality is an embedding property

I was reading Finite Soluble Groups by Doerk and Hawkes and they mention that pronormality is an embedding property. What is meant by this embedding property? and how is this related to pronormal ...
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0answers
10 views

conjugacy of invertible matrices over different fields [duplicate]

Let $K$ denote one of the fields $\mathbb{Q}$, $\mathbb{R}$ or a number field. Consider $A,B\in \mathrm{GL}_n(K)$. Question: Is it true: $A,B$ are conjugate in $\mathrm{GL}_n(\mathbb{C})$ $\implies$...
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0answers
22 views

amalgamated free product of von Neumann algebras

If $G$ and $H$ are two discrete groups and $L(G)$ and $L(H)$ be their group von Neumann algebras and $A$ be their common *-subalgebra, what can we say about their amalgamated free product under $A$, i....
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2answers
37 views

Can this be done using Sylow theorems? [duplicate]

Let $p$ and $q$ be distinct primes.Suppose that $H$ is a proper subset of the integers and $H $is a group under addition that contains exactly three elements of the set {$p,p+q,pq,p^q,q^p$}.Determine ...
0
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3answers
67 views

On a property of generating set for groups

Consider the following paragraph from Lang's Algebra: My question is about converse of one statement, which I was not able to prove. Question: If $f(S)$ do not generates $F$, then can we find ...
3
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1answer
68 views

What is significance of this proof of existence of free groups (Lang's Algebra)

There are different proofs of existence of free groups. While reading Lang's Algebra, it caught my attention towards proof of this theorem by first bracket statement in proof: Later I went on ...
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1answer
58 views

Need help in understanding the proof of “If $ \vert G \vert$=60 and $ G $ has more than one Sylow 5-subgroup, then $ G $ is simple.”

This proof is from Dummit & Foote text. Suppose by way of contradiction that $\vert G \vert=60$ and $n_5$>1 but that there exists $H$ a normal subgroup of $G$ with $H$ $\neq$ $1$ or $G$. By Sylow'...
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0answers
33 views

What kind of map is the inversion map on groups?

Let $G$ and $H$ be groups. Suppose $\phi : G \rightarrow H$ is a map such that $\phi(g g') = \phi(g') \phi(g)$ for all $g,g' \in G$. What is the name for such a map? For example, if $H = G$, then $\...
2
votes
1answer
25 views

Commensurability for vector spaces

Let me start by saying I am a not a mathematician and I have not studied group theory (just a few brushes here and there) but after reading I have a very basic understanding of commensurability as ...
2
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1answer
41 views

Determining a group given some elements

Say we have a group G and we know some of the elements (but not all). How does one determine the order and list all the elements of the group in an intuitive way? In this case G is the smallest ...
0
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2answers
35 views

Order of factor group

Question: Determine the order of $(\mathbb{Z} \times \mathbb{Z})/ \left<(4,2)\right>$. Is the group cyclic? I want to first apologize for the way this post is written. I'm on the road and ...
2
votes
3answers
89 views

Subgroup of Order $n^2-1$ in Symmetric Group $S_n$ when $n=5, 11, 71$

$G$ is a symmetric subgroup of symmetric group $S_n$ acting on $n$ objects where $n=5, 11, 71$ and order of $G$ is $n^2-1$. Question: Does $G$ (as defined above for $n=5, 11, 71$) exist? How can I ...
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2answers
54 views

For a prime $p$ if $p^m = p^n+2\cdot p^k$ then $p=3$.

I read an article on commuting graphs of groups and at some point, author gets the equality $|\langle x,Z\rangle| = |Z|\cup 2\cdot |x^G|$ where $Z$ denotes the center of the $p$-group $G$ and $x^G$ ...
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0answers
13 views

Finite number of orbit type for compact Lie group actions

For the linear SO(3) actions, the number of orbit types (or the number of isotropy class) is finite : this seem to be a classical result coming from Bredon (Introduction to compact transformation ...
3
votes
2answers
90 views

Homomorphism between a group of exponent $m$ and $\mathbb{Z}/m\mathbb{Z}$

Let $G$ be an abelian group of exponent $m$, where $m\in\mathbb{N}$. Is there always a nontrivial group homomorphism between $G$ and $\mathbb{Z}/m\mathbb{Z}$ ? For example, if we have $G=\mathbb{Z}/m\...
0
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1answer
44 views

Does $\text{Gal}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ act on $\overline{\mathbb{Q}}$ with an infinite orbit?

Does $\text{Gal}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ act on $\overline{\mathbb{Q}}$ with an infinite orbit? I know the definition of orbit and I know that $\sigma$ in Galois group changes roots. I ...
1
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1answer
37 views

Pre-image of a join of subgroups

Let $\phi : G \longrightarrow H$ be a surjective homomorphism. Suppose that $W\leq H$ and let $x\in H$ such that $x= \phi(g)$ for some $g\in G$. I would think that the following relation holds, though ...
3
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0answers
49 views

How to find the smallest set of generating elements in a group?

Is there a systematic procedure for finding the smallest set of generating elements of a finite group?
1
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1answer
30 views

Counting subgroups of free product of copies of $\mathbb{Z}$ with certain index

For a natural number $n$, let $Z_n=\mathbb{Z} \ast \cdots \ast \mathbb{Z}$ denote the free product of $n$ copies of the integers. Let $m$ be a further integer. $\textit{Question:}$ Is there a ...
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22 views

Pre-image of conjugate subgroups

Let $\phi : G \longrightarrow H$ be an epimorphism. If $W$ and $K$ are cojugate in $H$ then $\phi^{-1}(W)$ and $\phi^{-1}(K)$ are conjugate in $G$. Firstly is this true for the above condtions? I ...
0
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0answers
24 views

Relation between permutation group and algebraic equation. [duplicate]

What kind of relation do algebraic equeation and permutation group have? For example, $Z^n -1=0$ is related to a cyclic group $C_n$. Is there anything else in this kind problem? I have read about ...
2
votes
1answer
33 views

Permutation of Disjoint Sets of a Symmetric Group

Problem Description: Consider a symmetric group $S_n$ acting on $n$ objects. We partition $S_n$ into two sets $A, B$ such that $A \cap B= \emptyset$ and $A \cup B = S_n$. In other words, $S_n$ is ...
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0answers
39 views

Creating a Group from another Group and Automorphism of that group?

Suppose $G$ is a group and $T$ is an automorphism of $G$ of order $k$. We create a group $\{G,T\}$ (Construction given in Herstein 2nd edition Pg 69). Now in the explanation given in text, it ...
3
votes
2answers
104 views

How to write the commutator subgroup in terms of the generators of the group?

Let $G=\langle\ S\ |\ R\ \rangle$ be a finitely presented group. The commutator subgroup of $G$ is the group generated by $\{[a,b]\ |\ a,b\in G\}$ and is denoted by $[G,G]$, where $[a,b]=aba^{-1}b^{-1}...
2
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3answers
91 views

If $G/N\cong H$ then $G=NH$?

I am curious if $G/N\cong H$ then $G=NH$? ($N$ is a normal subgroup of $G$, and $H$ is a subgroup of $G$.) With this setup, we get that $NH$ is a subgroup of $G$ so $NH\subset G$. I am not sure ...
3
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1answer
36 views

Two equivalent conditions proof (related to semiproduct of groups)

Let $G$ be a group, and let $N$ be a normal subgroup of $G$, $H$ any subgroup of $G$. I wish to prove the equivalence of (i) $G$ is the product of subgroups, $G=NH$, where $N\cap H=\{e\}$. (ii) ...
0
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1answer
41 views

solve a specific word problem in free groups

Let $F_2=\langle a, b\rangle$ be the non-abelian free group with two generators and $e$ is the neutral element in $F_2$. Given $g\in F_2, k\geq 2$ an integer. I want to know how to solve the word ...
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0answers
25 views

Sylow tower theorem involving supersolvable groups

I just want to find out if anyone has a reference to the result that states that if $G$ is a finite supersolvable group then it has a normal Sylow subgroup.
3
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1answer
25 views

Prove that L(x) = (x - 1)/p is a discrete logarithm function in a group.

I have the following problem:$$$$ Let $p$ be a prime number and $G$ be a set of all $x\in \mathbb{Z}_{p^2}$, such that $x \equiv 1 \pmod{p}$. Prove that: $G$ is a multiplicative group (regarding ...
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1answer
44 views

soft question- Kenneth's Brown notation at “Cohomology of finite Groups”

As the title indicates, my question has to do with something rather simple. So, in Kenneth's Brown book "Cohomology of finite groups" at pg.84-85 and in particular Theorem 10.3 and Proposition 10.4, ...
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1answer
25 views

Reference for results p-adic integers Z_p as abelian group

I have two facts I want to use in my thesis about $\mathbb{Z}_p$. To be precise: automorphism group is $\mathbb{Z}_p \times \mathbb{Z}/(p-1)\mathbb{Z}$, except for 2, and that any subgroup with finite ...
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23 views

Finding a 3-embedded subgroup.

I have the group of order 108 $G=(((\mathbb{Z}_3 \times \mathbb{Z}_3)\ltimes \mathbb{Z}_3)\ltimes \mathbb{Z}_2) \ltimes \mathbb{Z}_2$ obtained from an algorithm in GAP, but I need to prove that it has ...
1
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1answer
49 views

Product with a normal subgroup

If $H \unlhd K \unlhd G$ and $P\unlhd G$ then does necessarily $HP \unlhd KP$? I can see this is true using the correspondence theorem since $HP/P \unlhd KP/P$ I want to try direct and prove it ...
2
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0answers
64 views

Commensurability of two groups

If two groups $G\bigoplus \mathbb{Z}$ and $H\bigoplus \mathbb{Z}$ are commensurable. Does it imply that the groups $G$ and $H$ are commensuarble?
1
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1answer
26 views

Supersolvable and pronormal subgroups

Let $G$ be a finite group such that all subgroups of prime-power order are pronormal in $G$. If $M$ is a normal $p$-subgroup of $G$ then all prime-power order subgroups of $G/M$ are pronormal in $G/M$....