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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Algebraic Structures that do not respect isomorphism

One of the first things a student learn in Algebra is isomorphism, and it seems many objects in algebra are defined up to isomorphism. It then comes as a mild shock (at least to me) that quotient ...
2
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1answer
34 views

Properties of group extensions

There is an excercise in Rotman's Introduction to the theory of groups: which of the following properties, when enjoyed by both $K$ and $Q$, is also enjoyed by every extension of $K$ by $Q$? finite $...
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1answer
128 views

A group is isomorphic to the direct product of two subgroups. Conditions for the subgroups to be normal.

Let $(G, \star, \varepsilon)$ be a group and $H$ and $K$ two of its subgroups and let $$\begin{align*} \diamond\, \colon (H \times K)^2 &\to H \times K \\ \big((h_1,k_1) , (h_2,k_2)\big) & \...
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1answer
37 views

Monomorphisms, epimorphisms and isomorphisms of groups category

I would like to solve the Problem 2.12 in Szekeres, A Course in Modern Mathematical Physics: Show that the class of groups as objects with homomorphisms between groups as morphisms forms a ...
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1answer
35 views

Showing a group is not the internal direct product of cyclic group

Question: Let$ G={3^{a}6^{b}10^{c}}$ under multiplication and $H={3^{a}6^{b}12^{c}}$ For all $a,b,c \in R$ Prove that$ G=<3>x <6>x <10> $ and H is not $<3>x <6>x &...
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1answer
22 views

How to save a pc-group in MAGMA?

I have a group $M$, which is a large matrix group. Using the command $\texttt{LMGSolubleRadical(M)}$ I obtained $R\cong M$, where $R$ is of type $\texttt{Grppc}$, and an isomorphism $M\to R$ called $\...
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
37 views

Infinite length Composition series

Let $G$ be a group (possibly infinite). Suppose $G$ has a composition series. I could show that any other composition series has the same length. But I cannot prove the following. Let $G \...
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2answers
36 views

Cardinality of a vector space over a finite field [closed]

Let $V$ be a vector space over $\mathbb{Z}_5$ of dimension $3$. What is the cardinality of $V$? I don't know how to proceed. Thanks in advance.
3
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0answers
81 views

Is this equation $(n+1)~x^{2n+1}-n~x^{2n}-n=0$ solvable in radicals for some $n \geq 2$?

Consider this polynomial equation: $$(n+1)~x^{2n+1}-n~x^{2n}-n=0,~~~~n \geq 2,~~~n \in \mathbb{N}$$ It's related to another question of mine, but I don't think the context matters here. I'm ...
0
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1answer
21 views

Relationship between elements in an internal direct product

Question: Let $H$ and $K$ be subgroups of a group $G$. If $G=HK$ and $g=h\bar {k} $, where $h\in G $ and $\bar {k} $. Is there any relationship among $|g|$, $|h|$ and $|\bar {k}|$? What if $...
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1answer
16 views

Express a group in four different direct product

Question: Express U (165) as an internal dirext product of proper subgroup in four different ways. The hint suggests the use of the Chinese remainder theorem which I am unfamiliar with. Is there ...
0
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1answer
29 views

Find the Quotient group of normal subgroup of order 4 of quaternion group.

$H=(\pm 1 , \pm i , \pm j, \pm k)$. I know $$c(i)=( \pm 1, \pm i) \quad c(j)=( \pm 1, \pm j)\quad c(k)=( \pm 1, \pm k) $$ and thus the class equation of $H$ is $8= 2+2+2+2$ because $\lvert H\rvert =...
4
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2answers
106 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 $\exp^{-1}$, ...
4
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1answer
35 views

Left vs right semi direct products

I just want to make sure that I am not doing anything silly here, but if we let $G$ be a group with $H,K$ subgroups, $H\lhd G$, and $\phi:K\rightarrow Aut(H)$, then is $$H\rtimes_\phi K \approx K \ _\...
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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 ...
0
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2answers
37 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 ...
2
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1answer
30 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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50 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 ...
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0answers
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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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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 ...
3
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1answer
99 views

A small cancellation group does not contain $\mathbb{Z}^3$

I read somewhere that a small cancellation group (ie. a group admitting a presentation statisfying the small cancellation condition $C'(1/6)$) does not contain $\mathbb{Z}^3$, but without a precise ...
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24 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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0answers
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 ...
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 $n=5,11,71$, ...
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 ...
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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'...
10
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1answer
180 views

Central Quotients of Finite Groups

There are more than 50 groups of order 48, and among them 16 groups have center of order 2, let $G$ be among such groups. Then $G/Z(G)$ is a group of order 24. What is this group of order 24? There ...
3
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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\...
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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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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 $\mathrm{GL}_n(\mathbb{C})$ $\implies$...
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 ...
0
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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 ...
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 ...
3
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1answer
40 views

Why is a nontrivial finite group is nilpotent if every maximal subgroup is normal?

In the following proof I understand the above proof up to the part where it says "by Sylow theory $N(M)=M$", could someone explain to me why is this true. We have just started learning about group ...
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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
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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 ...
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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
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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 ...
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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
14 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 ...
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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 ...
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 ...
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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?
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1answer
469 views

Proof that $U(n)$ is connected

I'm trying to prove that $U(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I\}$ is connected, but most of the proof comes down to proving that $SU(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I $ and $ detX=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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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 ...
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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 ...