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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An automorphism on generating set of a group

Let $G$ be a finite group and $A=\{a_{1},...a_{k}\}$ and $B=\{b_{1},...,b_{k}\}$ be two minimal generating sets of $G$ such that $|a_{i}| = |b_{i}|$ for $i=1,\dots,k$. We define ...
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On automorphisms of finite abelian group

Let $G$ be a finite abelian group such that $a, b\in G$ and $\mid a\mid=\mid b\mid$. Then does there exist an automorphism of $G$ such that $\alpha(a)=(b)$? Thank you
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260 views

The group of all bijections

Let $X$ be a set and let $Y \subset X$ be a subset with $y\in Y$. Let $S_X$ be the group of all bijections from $X$ to $X$. I am given four sets, and I am to determine if each is a subgroup of $S_X$. ...
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439 views

On which structures does the free group 'naturally' act?

One of the best ways to get a handle on a group is to recognize it as isomorphic to a set of symmetries of some structure. The dihedral group of order $2n$ is easily recognized as the set of ...
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1answer
110 views

On a theorem of Burnside

The following is a well known theorem of Burnside, which I am reading from the original paper. If $G$ is a finite $p$-group and $\{S_1,S_2,\cdots, S_n\}$ is a set of elements of $G$ such that their ...
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63 views

Find an isomorphism for this map

Suppose $G = (\mathbb{R}^*,\cdot)$ is the multiplicative group of nonzero real numbers. $\mathbb{Z}_2=\{0,1\}$ has operation $0+0=0, 0+1=1, 1+0=1, 1+1=0$ and is a group. Let $\mathbb{R}$ be the group ...
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1answer
130 views

Subgroup of an abelian Group

I think I have the proof correct, but my group theory is not that strong yet. If there is anything I am missing I would appreciate you pointing it out. Let $G$ be an abelian group (s.t. $gh = hg$ ...
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1answer
424 views

Generators of Translation - Lie Algebra [duplicate]

I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from $(x,y)$ to $(x+a,y+b)$ then can we write it as : $$ \left( \begin{array}{ccc} ...
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603 views

Translations in two dimensions - Group theory

I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from $(x,y)$ to $(x+a,y+b)$ then can we write it as : $$ \left( \begin{array}{ccc} ...
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2answers
87 views

What are the Subgroups of $\Bbb{Z}_2 \times \Bbb{Z}_4$?

I know that if $Z_n$ is cyclic, then every subgroup of $Z_n$ is also cyclic, and every such subgroup is generated by a single element. However, $\Bbb{Z}_2 \times \Bbb{Z}_4$ is clearly not cyclic, ...
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97 views

Basic Abstract Algebra - Homomorphism [duplicate]

Given a homomorphism $f:G \rightarrow H$, $G$ finitely generated, what can you say about the order of $g_i$ and $f(g_i)$? I've thought about this question for a while but haven't come to a ...
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1k views

Proving associativity

consider $G = \mathbb{R}_{\geq 0}$ with binary operation $\star$ s.t. $a \star b = |b-a|$ my attempt; $(a\star b)\star c = |c - a\star b| = |c - |b-a||$ $a\star (b\star c) = |(b \star c) - a| = ...
3
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1answer
437 views

Groups of order $pq$ have a proper normal subgroup

I am doing the following exercise from [Birkhoff and MacLane, A survey of modern algebra]: Let $G$ be a group of order $pq$ ($p,q$ primes). Show that either $G$ is cyclic or contains an element ...
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Abelianization of free group is the free abelian group

How does one prove that if $X$ is a set, then the abelianization of the free group $FX$ on $X$ is the free abelian group on $X$?
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1answer
124 views

An interesting question from “Group Theory: A First Journey”

I am currently studying the manuscript Group Theory: A First Journey by Vipul Naik. It is available from the web page http://www.cmi.ac.in/~vipul/mathjourneys/ . In this manuscript the author proposes ...
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What are the subgroups of $C_2\times C_{202}$?

Basically, if $C_2$ is a cyclic group of order two and $C_{202}$ is the same with order $202$, what are the subgroups of the product of the two? The farthest I've gotten is that if $C_2$ is addition ...
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177 views

Why isn't $\langle a ; a^2 \rangle$ (or $\langle a;a^3, a^7\rangle$) a presentation of $C_4$?

I've just read the first few pages of Combinatorial Group Theory by Magnus, Karrass, and Solitar, and based on their definitions there, and more specifically, the reasoning given in the hint to ...
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1answer
152 views

elementary abelian groups and finite fields

in group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order $p$, where $p$ is a prime; it is a particular kind of $p$-group. now suppose that we ...
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2answers
65 views

Prove for $ar^k = r^{-k}a$ all integers $k$

Please help. I am working with the symmetries of a square. Prove $ar^k = r^{-k}a$ for all integers $k$. Attempt: Base case: $P(1)$ is true since $ar = r^{-1}a$. Let $P(k)$ be the statement ...
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1answer
54 views

How do we compute $\mathbb{Z}^2/(n, m)$?

I have been trying to compute quotients of this form (as modules). Does anyone have a quick method of doing this? I'm sure I'm missing something obvious. Any help is appreciated!
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1answer
62 views

Problem of group theory

Let $G$ be a group under matrix addition (entries belong to $\mathbb Z$) and $H$ be a subgroup of $G$ consisting of matrices with even entries. Then how to find the order of quotient group $G/H$.
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2answers
266 views

Is the derivative function a group homomorphism on $G$?

Let $G$ be the set of real-valued, differentiable functions (the operation on $G$ is addition). Is the derivative function $D$ a group homomorphism from $G\rightarrow G$? I'm pretty sure it is. Let ...
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3answers
100 views

Generating Alternating Groups

Is there a way to think about how to generate alternating groups? Say I wanted to generate the alternating groups $A_3,A_4,A_5$.
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2answers
119 views

Basic Abstract Algebra - Subgroups of Abelian Group

I'm trying to prove the following: Let $G$ be an abelian group of order 72. Show that $G$ has exactly one subgroup of order 8. I know by theorem that $G$ must have at least one subgroup of order ...
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1answer
49 views

Elements of a group

Consider the following presentation for $A_4$ $$< p, q\, |\, p^2 = (pq)^3 = q^3 =e>$$ There exist eight elements of order $3$ in this group. Deduce these elements by writing them as products ...
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79 views

Set of Normal subgroups is a sublattice of a set of subgroups

I need to show that if $ G$ is a group then $\mathcal N(G)$ is a sublattice of $S(G)$. Obviously $N(G) \subseteq S(G) $. How do I show that operations join and meet agree with those of the original ...
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208 views

On the centralizers of $n$-cycles and conjugacy in $A_n$

I'd appreciate comments on the validity of these attempted proofs. Thanks. Let $a$ be an $n$-cycle in $S_n$. a) Show that the centralizer of $a$ in $S_n$ is $\langle a \rangle$. b) Assume that $n$ ...
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1answer
117 views

understanding into algebraic terms difference between homology and cohomology

my previous question understand quotient group was related to understanding of quotient group,i dont need to know too much detailed in group theore,just some part of algebraic topology,especially ...
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2answers
132 views

understand quotient group

i am trying to understand what does mean quotient terminology in group theory by as simple way as possible,also quotient group i want to know something about it,using internet i read that " In ...
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188 views

The center of a group with order $p^2$ is not trivial

Let $p$ be a prime and $G$ be a group of order $p^2$. Show that $Z(G)\neq 1$. Is there a proof of this nice fact that doesn't use the class equation?
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finite normal subgroup

$G$ is a subgroup of finite index in $SL(n,Z)$, $n\ge 3$, $N$ is finite normal subgroup of $G$, then I want to know why $N$ is a normal subgroup of $SL(n,Z)$. More generally, $A$ is an arithmetic ...
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Intuition - Fundamental Homomorphism Theorem - Fraleigh p. 139, 136

Let $\phi: G \to H$ be a group homomorphism with $K = \ker\phi$. Then $G/K \simeq \phi[G]. $ The hinge to the proof is to define $\Phi: G/K \to \phi[G]$ given by $\Phi(gK) = \phi(g)$. Then we must ...
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Every element of a finite abelian group with square free order equivalence

I'm currently having some trouble with this problem: Given $G$ a finite abelian group, prove the following are equivalent: $1.$ Given any subgroup $H$, there exists a subgroup $K$ ...
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1answer
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Examples of arbitrary groups

If $G$ is an arbitrary group with order $m$ and $n$ divides $m$, then $G$ need not have a cyclic subgroup of order $n$. Find two such examples, one where $n=m$ and another where $n<m$. Should I ...
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381 views

Centralizer of involutions in simple groups.

I need some information about centralizer of involutions in finite simple groups of lie type. Actually I want to know if $G$ is a simple group of lie type over a finite field, How many conjugacy ...
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191 views

If $G$ is a finite group where every non-identity element is generator of $G$, what is the order of $G$?

If $G$ is a finite group where every non-identity element is generator of $G$, what is the order of $G$? I know that the order of $G$ must be prime, but I'm not sure how to go about proving this ...
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1answer
21 views

Function with some property

A function $f$ is defined on the set $\{0,1,2,3,…,n-1\}$ to itself. This is such a function that if you take any $k$ from the set $\{0,1,2,3,…,n-1\}$ then $f^m (k)=0$ for some natural number $m$. ...
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Would the transformation of a differential equation obey the same algebra?

I've found that the algebra of this differential equation $$\frac{d^2y}{dz^2}-(3z^2+\gamma)\frac{dy}{dz}+(cz+\alpha)y=0$$ is in $sl(2)$ because it is possible to use the generators of the $sl(2)$ ...
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3answers
115 views

Solutions of $x^2=1$

Show that in a cyclic group, the equation $x^2=1$ has no more than two solutions. Since $G$ is a cyclic group, an element $x\in G$ can be written as $\langle x \rangle=\{x^n\mid ...
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1answer
105 views

Find a homomorphism that is also a bijection

Question: Let $G=(\mathbb{R}^{*},\cdot)$, be the multiplicative group of nonzero real numbers. Recall that $\mathbb{Z}_{2}=\{0,1\}$ with the operation $0+0=0$, $0+1=1$, $1+0=0$, $1+1=0$ is a group. ...
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2answers
490 views

Relatively prime orders of cyclic groups

If $a$ and $b$ are elements of a group whose orders are relatively prime, what can you say about $\langle a\rangle\cap \langle b\rangle$? Let the order of $a$ be $m$ and the order of $b$ be $n$. ...
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1answer
133 views

Field Extensions and their dimensions

There is a powerful theorem with respect to a field $F$ extensions and their dimensions. $F<E<K \ \Rightarrow [K:F] = [K:E][E:F] $ This is analogous to the famous Lagrange's theorem with ...
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Counting simple quadrilaterals in a rectangular lattice.

I've been trying to make an algorithm to find the number of all possible simple quadrilaterals in a N*M lattice. I already have a brute force solution but since this is a Project Euler problem I ...
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2answers
62 views

Why does this inverse work

$U_n$ is defined to be a set such that $ \{\ [a] \in \mathbb{Z_n} \mid \gcd(a,n)=1 \}\ $ where $ \{\ [0] \}\ $ is obviously omitted. I'm proving that $(U_n , \cdot_n)$ is a group and I have come to ...
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3answers
503 views

How do I draw the lattice of subgroups for this group?

Question: The symmetry group of a regular pentagon is a group of order 10. Show that it has subgroups of each of the orders allowed by Lagrange's theorem, and sketch the lattice of subgroups. I got ...
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1answer
76 views

automorphism of fields

let we consider $GF(p^n)$ as a vector space over $GF(p)$, $p$ is prime. Also we want to have an invertible linear map on $GF(p^n)$, (automorphism of $GF(p^n)$). on the other hand we know that A field ...
2
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1answer
47 views

cyclic group definiton

i would like to clarify some things related to group theory,for example let us consider two group,one with modulo $3$ and second with modulo $2$,so we have $G_3=(0,1,2)$ and $h_2=(0,1)$ now if i ...
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1answer
393 views

Free product of groups as coproduct

Wikipedia says "The coproduct in the category of sets is simply the disjoint union with the maps ij being the inclusion maps. Unlike direct products, coproducts in other categories are not all ...
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448 views

Questions on Proofs - Equivalent Conditions of Normal Subgroup - Fraleigh p. 141 Theorem 14.13

(1.) Why did Fraleigh shirk the proof for $(2) \implies (1)$? By dint of Arthur's comment, $(2) \iff \color{crimson}{gHg^{-1} \subseteq H} \quad \wedge \quad gHg^{-1} \supseteq H \implies ...
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1answer
221 views

Tricks, Multiply across Subset - Left Coset Multiplication iff Normal - Fraleigh p. 138 Theorem 14.4

Left coset multiplication is well defined by $(aH)(bH) = (ab)H \iff H \triangle G$. Given $H\leq G$, we wish to define a group structure on $G/H$ under suitable conditions. The natural way to do ...