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.

learn more… | top users | synonyms (2)

0
votes
1answer
145 views

Do the set of functions mapping real numbers to real numbers form a group under composition?

Consider the set of functions $\left\lbrace f: \mathbb R \to \mathbb R\right\rbrace$. Define $f_1 \star f_2$ to be the composition $f_1 \circ f_2$. With the operation $\star$ is this set of functions ...
1
vote
4answers
451 views

Show that if $ab$ has finite order $n$, then $ba$ also has order $n$. - Fraleigh p. 47 6.46.

This solution is from here and yahoo. Given $a,b$ elements of $G$, and $ab$ has finite order $n$. Hence $\color{magenta}{|ab| = n} \iff (ab)^n = e$. Need to show $n$ is the smallest positive integer ...
33
votes
5answers
2k views

Can every group be represented by a group of matrices?

Can every group be represented by a group of matrices? Or are there any counterexamples? Is it possible to prove this from the group axioms?
0
votes
2answers
73 views

Injective Homomorphism Between a Group and its Permutation

I've got the following two-part question. I fiddled with it for a while, but didn't figure it out. Any help would be appreciated. Let $G$ be a group, and $g \in G$. Denote by $R_g: G\to G$ the map ...
1
vote
2answers
195 views

Cancellation property in groups

Prove that the following ‘cancellation property’ holds in any group: ab = ac implies b = c, and ba = ca implies b = c. Need help with this, don't know how to prove it for a group.
2
votes
1answer
114 views

Showing that the diagonal of $G \times G$ is maximal, where $G$ is simple

I have been trying to prove the following: Let $G$ be simple, and write $\Gamma=G \times G$. Let $D \le \Gamma$ be the diagonal subgroup, which consists of all elements of the form $(x,x)$, where $x ...
2
votes
1answer
55 views

Abstract Algebra: Index of Subgroups

Here's the problem I'm working on: Prove: Suppose $H$ has index $p$ and $K$ has index $q$, where $p$ and $q$ are distinct primes. Then the index of $H \cap K$ is a multiple of $pq$. (Plus: do you ...
4
votes
1answer
324 views

A game of Chess - Ideal Solution

I am a student of physics. I have learnt some basic group theory, and I am wondering if there is any ideal solution for a given Chess game (like solving Rubik's cube). I know the no. of permutations ...
4
votes
1answer
502 views

Group question: only one element $x$ with order $n>1$, then $x\in Z(G)$

I just tried to solve this question on a exam: Let $G$ be a group with only one element $x$ of order $n$, $n$ natural. Show that $x\in Z(G)$. (I'm using multiplicative notation) $Z(G)$ is the ...
2
votes
1answer
49 views

find a special element of $Sp(2n,q)$

Please help me about this question: I want to show that symplectic group $Sp(2n,q)$ has an element of order $q^n-1$.
1
vote
1answer
134 views

Subdirect product of simple groups

Let $G$ be a subdirect product of simple groups $G_i$ for $i\in I$. I want to show that if $1\neq x\in G$, then there exists a maximal normal subgroup $N$ such that $x\notin N$. First of all, I know ...
6
votes
2answers
176 views

Abelian groups whose subgroup lattice is chain

Let $G=Z(p^k)$ where $k=1,2,..,\infty$. The group $G$ has exactly one subgroup series. Does exist an other infinite abelian group with this property?
3
votes
2answers
145 views

Subgroup with Finite Index of Multiplicative Group of Field

Let $F$ be an infinite field such that $F^*$ is a torsion group. We know that $F^*$ is an Abelian group. So every subgroup of $F^*$ is a normal subgroup. My question: Does $F^*$ have a proper ...
4
votes
1answer
137 views

Frattini subgroup of $\operatorname{Hol}(\mathbb Q)$

Let $\mathbb Q$ be rational number under addtion. Is Frattini subgroup of $\operatorname{Hol}(\mathbb Q)$ trivial where $\operatorname{Hol}(\mathbb Q)=\mathbb Q \rtimes \operatorname{Aut}(\mathbb Q) ...
6
votes
2answers
108 views

Group with Subgroup of Index 2

I am looking group $G$ such that $G$ has exactly two subgroups of index 2. I have searched by GAP but I couldn't find it.
5
votes
2answers
197 views

Torsion subgroup of $SL(n,\mathbb Z)$

Let $G$ be subgroup of $SL(n,Z)$ such that for any $g\in G$ there exists integer $m\geq1$ $g^m=1$. Show that there exists $N\geq1$ such that for any $g\in G$ , $g^N=1$ I know $m$-th root of unity is ...
7
votes
1answer
849 views

order of element in symmetric group

let $n=p_1+p_2+\cdots+p_k$ ($p_k$ is kth prime number) then $\prod_{i=1}^k p_i$ is maximum order in $S_n$. I think it is easy but I am trying to prove it , but I have not any idea how to deal with ...
5
votes
1answer
466 views

$PGL(n, F)=PSL(n, F)$

$PGL(n, F)$ and $PSL(n, F)$ are equal if and only if every element of $F$ has an $nth$ root in $F$.($F$ is finite field) I can show that if $PGL(n, F)=PSL(n, F)$ then $|F|$ have to be even.I have ...
3
votes
2answers
89 views

$[G:H]\leq[H:K]$

Let $G$ be finite group and $H<G$. For every proper subgroup $K$, $[G:H]\leq[H:K]$. I want to prove $H$ is normal subgroup. I fixed $K:=g^{-1}Hg$ but this doesn't work. Can somebody advise me?
0
votes
1answer
68 views

Maximal subgroups and isomorphism

Let $G$ and $H$ be two groups and $\phi:G\rightarrow H$ be a homomorphism. Suppose that $M$ and $N$ are maximal subgroups of $G$ and $H$, respectively. Now, are subgroups $\phi(M)$ and ...
1
vote
1answer
41 views

$G$ has a copy of $\dfrac{G}{N}$

Let $G$ be group and $N$ is a normal subgroup of $G$. Does $G$ have a copy of $\dfrac{G}{N}$? I really know that it is not true. However, I have a solution but I think that there exists an error. ...
1
vote
1answer
56 views

Show Burnside's Lemma (Weighted)?

Let $G$ be a group of permutations of $X$, and let $I(x)$ be an expression which is constant on each orbit of $G$, so that $$I(g(x))=I(x) \text{ for all } g \in G, x \in X.$$ Let $D$ be a set of ...
1
vote
0answers
30 views

From linear invariants of group to general ones

There is a lot of information about classical/linear invariants of finite groups. But does it lead to general invariants of group (for example, when we consider some action of our group on finite ...
-1
votes
1answer
32 views

Subgroup of order 24 in $S_7$?

Is there a subgroup of order 24 in $S_7$? What can we do to answer this question?
-1
votes
1answer
50 views

If $\phi[H] \subseteq H'$, homomorphism from G to G' induces homomorphism from G/H to G'/H' - Fraleigh p. 143 14.39

Let $H \trianglelefteq \text{ group } G$ and let $H' \trianglelefteq \text{ group } G'$. Let $\phi$ be a homomorphism of G into G'. Show that if $\phi[H] \subseteq H'$, then $\phi$ induces a natural ...
1
vote
1answer
66 views

Every cyclic subgroup of $F_n$ (the free group of rank n ) is separable

wondering how does one show that every cyclic subgroup of $F_n$ (the free group of rank n ) is separable?
2
votes
1answer
60 views

In the transition from set theory to order theory, what is the appropriate generalization of “group”?

Given a set $X$, the collection of all bijective endofunctions on $X$ forms a group, called the symmetric group on $X.$ Furthermore, Cayley's theorems says that every group embeds into a subgroup of ...
6
votes
2answers
160 views

Why is studying maximal subgroups useful?

When looking at finite group theory research, it seems to me that a lot of energy is devoted to determining the maximal subgroups of certain classes of groups. For example, the O'Nan Scott theorem ...
2
votes
2answers
72 views

Residually finite group with finitely many conjugacy classes of elements of finite order

Let $G$ be a residually finite group. Show that if $G$ has finitely many conjugacy classes of elements of finite order then $G$ has a torsion free finite index subgroup. Not sure how to get started ...
0
votes
3answers
69 views

How many subgroups there is for $\mathbb{Z}_{23}\times \mathbb{Z}_{31}$?

How many subgroups there is for $\mathbb{Z}_{23}\times \mathbb{Z}_{31}$? I try to understand how to solve it but I don't finding a way... I'll be glad if you help me with this... Thank you!
0
votes
1answer
236 views

Group isomorphism mapping generators to generators

I know that an isomorphism between cyclic groups maps generators to generators, but is this still the case if the groups are non-cyclic? Many thanks!
1
vote
3answers
66 views

Isomorphic groups between 2 sets

Show that the group $C_4 = \{ i, -1, -i, 1\}$ of fourth roots of unity in the complex numbers isomorphic to $\mathbb{Z}_4$. Can anyone please help me start this problem? I know that $C_4$ and ...
1
vote
1answer
32 views

Each vertex of this tree is either red or blue. How many possible trees are there?

The question: Let $X$ denote the set of 'coloured trees' which result when each vertex of the tree is assigned one of the colours red or blue. How many different coloured trees of this kind are there? ...
-1
votes
1answer
63 views

Finite groups with no common prime factor of their orders

Let $G$ and $H$ be finite groups s.t. their orders have no common prime factor, and let $\phi: G\rightarrow H$ be a homomorphism. I want to show that $\phi(g)=e \space \forall g \in G$ where $e$ is ...
2
votes
2answers
67 views

Let $G$ be a group, $H$ be a subgroup of $G$ and $h$ be a fixed element of $G$. [duplicate]

Let $G$ be a group, $H$ be a subgroup of $G$ and $h$ be a fixed element of $G$. Show that the subset $gHg^{-1}=\{ghg^{-1}:h\in H\} $ is a subgroup of $G$. I know of the one and two-step tests for ...
1
vote
1answer
59 views

Can Haar measure fail to be bi-invariant without conjugation shrinking a set?

Let $\: \langle \hspace{-0.02 in}G,\hspace{-0.04 in}\cdot,\hspace{-0.04 in}\mathcal{T}\hspace{.02 in}\rangle \:$ be a locally compact Hausdorff topological group, let $\mu$ be a left Haar measure on ...
2
votes
1answer
32 views

Properties of Groups

For any two elements x and y in G there exists z in G such that y=xz Prove it is true for every group or give counter example. So far I have: $ y=xz $ Multiplying on the left by $x^{-1},$ $ ...
2
votes
1answer
280 views

Conjugacy classes of elements of a prime order in $PSL_2(q)$

Let $q=p^f$ be a prime power. Given a prime number $r$, how many conjugacy classes of elements of order $r$ are there in $PSL_2(q)$? This topic should have appeared in literature, and I am told ...
2
votes
1answer
134 views

Why are the number of orbits considered the number of necklaces in this problem?

The question asks to show that there are just five different necklaces which can be constructed from five white beads and three black beads. There is a systematic example of how to solve this in the ...
5
votes
3answers
100 views

for every $\sigma\in \rm Aut(G)$ and for every $x\in G$, $\sigma(x)=x$ or $\sigma(x)=x^{-1}$

Let $G$ be a group such that for every $\sigma\in \rm Aut(G)$ and for every $x\in G$, $\sigma(x)=x$ or $\sigma(x)=x^{-1}$. I want to prove that $G$ is solvable.
2
votes
3answers
94 views

$n$-abelian Groups

Show that $(xy)^n=x^ny^n$ if $xy=yx$. I assume I will need 3 different cases: $n < 0$, $n=0$, and $n > 0$. For the $n > 0$ case, can I use induction? For the base case I'll show that ...
2
votes
1answer
78 views

Finiteness property of virtually torsion free groups

Do virtually torsion free groups always have finitely many conjugacy classes of finite subgroups? A paper I'm reading about $\mbox{Out}(F_n)$ mentions this finiteness property is a corollary of it ...
6
votes
2answers
197 views

What can we say about the kernel of $\phi: F_n \rightarrow S_k$

Let $F_n$ denote the free group on $n$ generators and let $S_k$ denote the symmetric group on the integers $\{1,\dots, k\}$, and the action of homomorphism $\phi$ (as given in the title) on the ...
3
votes
0answers
225 views

Commutator Identities in Groups

Let $x$, $y$, $z$ be elements of a group $G$ and let $[x,y]=x^{-1}y^{-1}xy$ be the commutator of $x$ and $y$. Then we have the following identities: $[x,zy]=[x,y][x,z][[x,z],y]$ ...
14
votes
1answer
390 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 ...
0
votes
1answer
132 views

Group Theory Exponent and Abelian Proof

Let G be a group such that $x, y \in G$ Show that, if $(xy)^2=x^2y^2$ or $(xy)^{-1}=x^{-1}y^{-1}$, then xy=yx. This can also be thought of as the exponent rule $(xy)^n$=$x^ny^n$ if xy=yx is true ...
3
votes
2answers
254 views

Group theory - Finite or Infinite group

I am just beginning to learn Group theory. As an example of finite groups our Professor wrote this group with just two elements, given by $$ \left( \begin{array}{cc} 0 & z \\ z^{-1} & ...
0
votes
2answers
26 views

Suppose k be a positive integer such that k divides p-1, where p is a prime. Prove that $\mathbb{Z}^*_p$ has an element of order k.

Suppose k be a positive integer such that k divides p-1, where p is a prime. Prove that $\mathbb{Z}^*_p$ has an element of order k. I think this is suppose to be the converse of Lagrange.
0
votes
2answers
80 views

Quotient group of a non-commutative group.

Let $Z(G)$ be the center of a non-commutative group $G$. Show that the factor group $G/Z(G)$ has at least 4 distinct subgroups. Attempt: So I have a guess at this however I feel that the answer is a ...
0
votes
1answer
42 views

What is the difference between $|3|$ and $|\langle 6\rangle|$ when finding the order?

What is the difference between $|3|$ and $|\langle 6\rangle|$ when finding the order? Is it asking the same thing with or without $\langle \rangle$? This is under $Z_{24}$. I know that $|\langle ...