0
votes
1answer
20 views

Prove that the order of an element in a cyclic group must divide the order of the group

Is it possible to prove this without invoking Lagrange's Theorem? Using it, the proof becomes trivially easy.
0
votes
1answer
28 views

How to think of group actions?

I am a little confused on how exactly I should be thinking of an action on a group. I have been trying to read up on it and came across Timothy Gower's blog which I think does a good job explaining ...
0
votes
2answers
30 views

Is the group $U(8)$ cyclic?

Referring to the group of units. My first thought is yes, since $1$ would be the generator. Although I think I'm getting confused between the generator and the identity element in this case. $1$ is ...
1
vote
2answers
29 views

finding all the left cosets

This is a homework problem: Determine if the index $[G:H]$ is finite or infinite. List all the left $H-$ cosets in the following: $a.$ Let $G=\mathbb{Z}\times \mathbb{Z}, \ H=\{(x,x):x\in ...
1
vote
1answer
39 views

Proof that the order of any finite $p$-group is a power of $p$

What is the most concise proof that the order of any finite $p$-group is a power of $p$?
-1
votes
1answer
32 views

Endomorphism and automorphism of S4 [on hold]

I'm stuck on how to calculating the Endomorphism and automorphism of S4. Thanks for your concerning.
1
vote
3answers
24 views

Lang Sylow Theorem

STATEMENT: Let $P$ be a $p$-Sylow subgroup of $G$ and H is a $p$-subgroup of G. Suppose first that $H$ is contained in the normalizer of $P$. We prove that $H\subseteq P$. Indeed, $HP$ is then a ...
0
votes
3answers
26 views

In what case the semi direct product is abelian?

Only when H$\rtimes$K is direct product and H,K are abelian? How to prove? In other words if the homomorphic from K to Aut(H) is not trivial then the semi direct product is not abelian?
0
votes
3answers
47 views

Subgroups of $S_4$ isomorpic to $S_2$

I am trying t prove that there are only 9 subgroups of $S_4$ isomorphic to $S_2$ but I only find 7 different ones( fix 12, 13, 14, 23, 24 34 and swap the remaining two + identity). Which cases am I ...
0
votes
2answers
33 views

Proving that $a$ is a $p$-cycle

I was reading Topic in Algebra by I.N. Herstein and trying to solve a problem from it. If $p$ is a prime number, show that in $S_p$ there are $(p-1)!+1$ elements $x$ satisfying $x^p=e$. I was ...
0
votes
1answer
15 views

Number of elements of order 2 in Z_60*Z_45*Z_12*Z_36

What is the number of elements of order 2 in Z_60*Z_45*Z_12*Z_36? Are there any short formula to find the number of elements of a given order in a group of direct product of some groups?
-1
votes
0answers
14 views

On Finite Monoid [duplicate]

Consider a finite monoid $(M,*)$. Let the identity element is the only idempotent element in $M$. Prove that $(M,*)$ is a group.
2
votes
0answers
33 views

Equivalence of categories $\Pi_1:\mathbf{K^1_{CW,*}}\to \mathbf{Grp}$

Let $K(G,1)$ denote the Eilenberg-Maclane spaces with fundamental groups isomorphic to $G$. Consider the category $\mathbf{K^1_{CW,*}}$ where objects are pointed $K(G,1)$ CW complex, and morphisms ...
0
votes
2answers
46 views

If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$ [duplicate]

I would like to prove that : If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$ I appreciate any clear explanation of this. Thanks !
2
votes
3answers
26 views

GCD's and how they generate groups

I was reading something today an it was talking about $U_{15}$, all the integers relatively prime to 15, and how it was generated by the set {7,11}. I understood it all, but I thought that if the ...
1
vote
1answer
44 views

Why $( Z_3\rtimes Z_2)\times Z_2 \cong (Z_3\times Z_2)\rtimes Z_2$?

I got an explanation, it says as $Z_2$ is in the kernel of the homomorphism. But I can't understand from that. Also can you tell me why $Z_3\rtimes Z_2\cong S_3$ ? Thank you.
0
votes
2answers
68 views

Does a Group being Finite Imply that It Is Cyclic?

I have been studying Abstract Algebra, and all the finite groups that we have studied so far have also been cyclic. So, is it true that all finite groups are cyclic? If yes, what is the theorem? If ...
0
votes
1answer
38 views

Defining an isomorphism

If we have to prove that the multiplicative group of integers modulo $8$, $U(8)$, is isomorphic to a set of matrices, are we allowed to define the isomorphism by saying: $$\begin{align} ...
0
votes
1answer
19 views

Introduction to Reidemeister--Schreier Method

I am learning Reidemeister--Schreier method, a method determining explicitly presentations for subgroups of a given group. Can anyone recommend some introductory material, preferably those with ...
1
vote
1answer
60 views

The Diagonal Subgroup of $A \times A$ is Maximal iff $A$ is Simple

Let $A$ be a group and $G = A \times A$. Define $D= \{(a,a,)\mid a \in A\}$ (the diagonal subgroup of $G$). Prove that $D$ is a maximal subgroup of $G$ if and only if $A$ is simple, i.e. it has no ...
0
votes
2answers
25 views

Find the number of prime ideals?(CSIR 2014)

Let $p,q$ be distinct primes. Then (1) $\dfrac{\mathbb{Z}}{p^2q}$ has exactly 3 distinct ideals. (2) $\dfrac{\mathbb{Z}}{p^2q}$ has exactly 3 distinct prime ideals. (3) $\dfrac{\mathbb{Z}}{p^2q}$ ...
3
votes
1answer
40 views

Bounding the order of a group by its nilpotentizer

Let $G$ be a finite non-nilpotent group. We put $nil_G(x)=\{y\in G\mid \langle x,y \rangle \text{ is nilpotent}\}$, called the nilpotentizer of $x$. Note that $nil_G(x)$ may not be a subgroup of $G$, ...
0
votes
2answers
42 views

Prove that if $f : \mathbb Z_p \to G$ is a homomorphism, then $f$ is either injective or trivial(i.e. $f(x)=1$ for all x).

I'm stuck on the last part that I assume there is one element other than $0$ and $1$ belongs to the kernel and I try to prove that $f(1)=1$, but I didn't see any clue to do that.
4
votes
1answer
51 views

Ways to find the order of an element in a group

Is there a better way of finding the order of an element in a group other than circling until the identity is reached? Is there or CAN there be a better general ways of finding orders of elements? ...
0
votes
5answers
73 views

Is it true, $O(ab)=O(ba),$ Where $G$ is a group and $a,b \in G.$

Suppose $O(a)$ and $O(b)$ is finite and also $O(ab)$ and $O(ba)$ is finite. Then L.C.M $(|a|,|b|)= L.C.M (|b|,|a|).$ (Is that Correct ?) Suppose $O(a)$ and $O(b)$ is finite but $O(ab)$ is infinite, ...
2
votes
0answers
36 views

If $G_1\cong G_2$ and $H_1\cong H_2$ then $G_1 \times H_1 \cong G_2 \times H_2$

If $G_1\cong G_2$ and $H_1\cong H_2$ then $G_1 \times H_1 \cong G_2 \times H_2$ Proof: $f_G:G_1\rightarrow G_2$ and $f_H:H_1\rightarrow H_2$. Question 1: Is the following statement valid? Does ...
0
votes
0answers
35 views

For a general group what does the formula $\sum_{g\in G} |G|/ord(g)$ mean? [on hold]

In my research, I have come across this group formula, $\displaystyle \sum_{g\in G} \frac{|G|}{ord(g)}$. Has anyone seen this before? Where have you seen this formula before? I am wondering if ...
2
votes
1answer
63 views

If $G,H$ are groups then $G\times H\cong H\times G$

This seems like a basic question, but I searched for a while and couldn't find it on the site. I want to know if I have a valid proof for the following theorem. If it is correct, I'd like to see how ...
0
votes
0answers
38 views

Finding an isomorphism from $\mathbb{R}^\times$ to a defined group $G$

Here's the problem I am solving: $G=\{x\in \mathbb{R}:x\not = 0\}$. The operation for $G$ is "$*$", with $x*y=\frac{1}{2}xy.\mathbb{R}^\times$ is the multiplicative group $\mathbb{R}.$ Find an ...
0
votes
2answers
20 views

Maximal normal subgroup has prime index

I am trying to solve the following exercise taken from Rotman's An Introduction to the Theory of Groups: Let $M$ be a maximal subgroup of $G$. Prove that if $M \lhd G$, then $[G:M]$ is finite and ...
-1
votes
0answers
25 views

Problem regarding the Lanczos Algorithm

I have a 8 by 8 matrix with entries: $(1,1) = 2\log(3) + 2\log(11) \equiv 4\log(2) + \log(5) \bmod{1008}$ $(3,3) = 2\log(5) + 2\log(7) \equiv 3\log(2) + 3\log(3) \bmod{1008}$ $(5,5) = 2\log(37) ...
0
votes
2answers
29 views

Cyclic Subgroup of Order 2

I came across something while looking up abstract algebra which said "Let G be a group and suppose there is an element $a$ in G which generates a cyclic subgroup of order 2 and is the unique such ...
1
vote
0answers
44 views

Solvability of word problem in group

I am fairly new to abstract algebra. So, I apologize if my question is too trivial. I am trying to prove that $G=\langle x,y \mid xy=yx; x^2=1\rangle$ has a solvable word problem. My idea is to show ...
0
votes
1answer
31 views

Proving that two quotient groups are isomorphic

Given a group isomorphism $\phi:G\rightarrow H$, and a subgroup $K \subset G$, I need to show that there is a group isomorphism $G/K\cong H/\phi(K)$, where $\phi(K)=\{h\in H \,\,\text{such ...
0
votes
0answers
20 views

Infinite Cyclic group representation

I am trying to learn Group representation and have a basic question regarding infinite cyclic groups. I am trying to find a representation of infinite cyclic group in $GL_n(\mathbb{C})$ and ...
-1
votes
0answers
18 views

The group $U_{34}$. Finding its subgroups. [duplicate]

I know U34=[1,3,5,7,9,11,13,15,19,21,23,25,27,29,31,33]. What are the proper subgroups? I know that there should be 4 of them: {1}, {$U_{34}$}. I just need to find the other 2, which will be of order ...
0
votes
1answer
39 views

An interesting puzzle for some, confusing for me

Suppose that $a$ is of odd order $k$ and $bab=a$. I need to show that $b$, must be of order $2$. We can prove this anyway we want to, but our hint is to expand $(bab)^k$ and re-associate and then ...
2
votes
2answers
24 views

Find the order of the elements in the given groups

I have to find the order of the following elements in the given groups: $(1 \ \ 2 \ \ 3) \ (1 \ \ 2\ \ 4) \text{ in } S_5$ $$\begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 3 & 4 & 1 ...
3
votes
0answers
144 views
+50

The correspondence theorem for groups

I had studied group theory a year ago, but still could not understand the proof involving The Correspondence theorem. let $G$ be a group and let $N⊴G$, where $N⊴G$ indicates that $N$ is a normal ...
0
votes
3answers
95 views

Is $\mathbb{Z}_7^*$ cyclic?

Determine whether the following sentence is correct or not. $$ \mathbb{Z}_7^* \text{ is cyclic. }$$ Is $\mathbb{Z}_7^*$ the same as $\mathbb{Z}$ without $0$?? If it is ...
5
votes
1answer
47 views

$p$ prime, Group of order $p^n$ is cyclic iff it is an abelian group having a unique subgroup of order $p$

I 've just read from An introduction to the theory of groups from Rotman's the following theorem: Theorem 2.19. Let $p$ be a prime. A group $G$ of order $p^n$ is cyclic if and only if it is an ...
1
vote
1answer
37 views

Free groups of rank greater than 2

I'm trying show that a free group of rank $\ge2$ is non abelian, but I have no idea to prove this. Any suggestions?
1
vote
1answer
39 views

Can we conclude from that, that there is an homomorphism between the group $G$ and the group $(\mathbb{Z}_3,+)$?

We have the third order group $G=\{1,g,x\}$, whose operation is the multiplication. To calculate the multiplication table we do the following: $1 \cdot 1=1, \ \ \ 1 \cdot g=g, \ \ \ 1 \cdot x=x$ ...
2
votes
2answers
48 views

Techniques for disproving group isomorphism

Suppose I wanted to find out if $f:\mathbb{Z}_6\rightarrow S_3$ is an isomorphism. Clearly, $f$ is bijective. It remains to show that $f(a+b)=f(a)\circ f(b)\;\forall a,b\in \mathbb{Z}_6$. For this ...
1
vote
1answer
33 views

Group of order $1575$ problem

Problem Let $G$ be a group with $|G|=1575$. If $H \lhd G$ and $|H|=9$, then $H \subset Z(G)$. What I've done so far is $|G|=1575=3^25^27$. I consider $G$ acting on $H$ by conjugation, or, in other ...
0
votes
1answer
21 views

Show that the groups are homomorphic

T={the n-th roots of unity} is a cyclic group of order n with the multiplication as operation. How can I show that there is a group homomorphism between this group and $(\mathbb{Z}_2,+)$ ?? Do I have ...
0
votes
0answers
23 views

Question about the third and fourth isomorphism theorems for groups

I am trying to work with both of the third and fourth isomorphism theorems for groups. I am considering the following situation: I wanted to take a subgroup in the quotient and see how it corresponds ...
1
vote
2answers
82 views

How does the index of this subgroup is a power of 2?

I am reading an article about coset codes (for answering this question having knowledge about these codes and lattice theory is not necessary) which are defined by $(\Lambda ,\Lambda ',C)$ in which ...
0
votes
1answer
41 views

A question in definition of group rings

In definition of a group ring $RG$ with elements $∑f_g g$ (where $g\in G$ and $f_g\in R$), are we supposed that $f_g$'s commute with $g$'s? I mean could we identify the above formal summation with $∑ ...
0
votes
1answer
34 views

classifying space of permutation groups

Given a group $G$, how to compute the classifying space of $G$? In particular, how to compute the classifying space $B\Sigma_n$ for a permutation group $\Sigma_n$?