Tagged Questions

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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.
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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 ...
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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 ...
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 $G=\mathbb{Z}\times \mathbb{Z}, \ H=\{(x,x):x\in ... 1answer 39 views Proof that the order of any finite$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$? 1answer 32 views Endomorphism and automorphism of S4 [on hold] I'm stuck on how to calculating the Endomorphism and automorphism of S4. Thanks for your concerning. 3answers 24 views Lang Sylow Theorem STATEMENT: Let$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$is then a ... 3answers 26 views In what case the semi direct product is abelian? Only when H$\rtimes$K is direct product and H,K are abelian? How to prove? In other words if the homomorphic from K to Aut(H) is not trivial then the semi direct product is not abelian? 3answers 47 views Subgroups of$S_4$isomorpic to$S_2$I am trying t prove that there are only 9 subgroups of$S_4$isomorphic to$S_2$but I only find 7 different ones( fix 12, 13, 14, 23, 24 34 and swap the remaining two + identity). Which cases am I ... 2answers 33 views Proving that$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$. I was ... 1answer 15 views Number of elements of order 2 in Z_60*Z_45*Z_12*Z_36 What is the number of elements of order 2 in Z_60*Z_45*Z_12*Z_36? Are there any short formula to find the number of elements of a given order in a group of direct product of some groups? 0answers 14 views On Finite Monoid [duplicate] Consider a finite monoid$(M,*)$. Let the identity element is the only idempotent element in$M$. Prove that$(M,*)$is a group. 0answers 33 views Equivalence of categories$\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)$CW complex, and morphisms ... 2answers 46 views If$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 ! 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 ... 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. 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 ... 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} ... 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 ... 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 ... 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} ... 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, ... 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. 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? ... 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, ... 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 ... 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 ... 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 ... 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 ... 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 ... 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) ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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. letG$be a group and let$N⊴G$, where$N⊴G$indicates that$N$is a normal ... 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 ... 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 ... 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? 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:$1 \cdot 1=1, \ \ \ 1 \cdot g=g, \ \ \ 1 \cdot x=x$... 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 ... 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 ... 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 ... 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 ... 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 ... 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$∑ ...
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 $\Sigma_n$?