0
votes
0answers
12 views

Prove that $U(m) = U_{m/n_1} (m) \times U_{m/n_1} (m) \times \cdots\times U_{m/n_k}$

If $m = n_1 n_2 \cdots n_k $ where $\gcd(n_i~,n_j)=1 ~~ \forall i \neq j$, then prove that: $$U(m) = U_{m/n_1} (m) \times U_{m/n_1} (m) \times \cdots\times U_{m/n_k}$$ where $\times$ refers to the ...
0
votes
0answers
37 views

Let $G$ be a finite abelian group and let $p$ be a prime that divides order of $G$. then $G$ has an element of order $p$

Let $G$ be a finite abelian group and let $p$ be a prime that divides order of $G$. then $G$ has an element of order $p$ Proof When $G$ is abelian. First note that if $|G|$ is prime, then $G \approx ...
3
votes
2answers
87 views

If the intersection of a normal subgroup and the derived group is {e}, show that N is a subset of Z(G).

I think my reasoning is wrong, but if the intersection only contains the identity, doesn't that imply that the only commutator in N is {e}, so doesn't that mean N is automatically commutative? Why was ...
2
votes
2answers
25 views

Supose that G is a finite abelian group that does not contain a subgroup isomorphic to $Z_p \oplus Z_p$ for any prime $p$. Prove that G is cyclic.

Supose that G is a finite abelian group that does not contain a subgroup isomorphic to $Z_p \oplus Z_p$ for any prime $p$. Prove that G is cyclic. Attempt: If $G$ is a finite abelian group, then let ...
1
vote
1answer
38 views

Can we give of the fact that a group of order $9$ is abelian without using an argument involving the product of two cyclic groups of order $3$?

A group of order $9$ is always abelian. I've seen proofs of this result, but I would like to prove it the following way: Let $G$ be a group of order $9$. If $G$ has an element $a$ of order $9$, then ...
0
votes
0answers
21 views

Identifying the abelian group with a presentation matrix

I am doing problems from Artin: \begin{bmatrix} 2 \\ 1\\ \end{bmatrix} and \begin{bmatrix} 2 & 4\\ 1 & 4\\ \end{bmatrix} For the First one after manipulating rows I ...
2
votes
1answer
51 views

Isomorphisms between finite abelian groups and cyclic groups

If G is abelian of order 175 and H is cyclic of order 25 and there is a homomorphism from G onto H then what is G isomorphic to? I can see how G is isomorphic to either $C_{25} * C_7$ or to $C_5 * ...
1
vote
0answers
50 views

Prove that a(mn)=a(m)a(n), (n,m)=1

Given a positive integer $n$ where $a(n)$ is the number of non-isomorphic abelian groups of order n. 1) Prove that $a(mn)=a(m)a(n), (n,m)=1$ 2) Prove that $a(p^k)$ is the number of partitions of k, ...
2
votes
1answer
20 views

if $G' <H < G$ then $H$ is normal in $G$.

if $G' <H < G$ then $H$ is normal in $G$. ($G'$ is the commutator subgroup of $G$.) This is what I do: because $G' < H$ we have $\frac{H}{G'} \triangleleft \frac{G}{G'}$. because ...
2
votes
2answers
79 views

How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?

I have been trying to calculate the number of subgroups of the direct cross product $\Bbb{Z}_4 \times \Bbb{Z}_6.$ Using Goursat's Theorem, I can calculate 16. Here's the info: Goursat's Theorem: Let ...
4
votes
1answer
121 views

Subgroups of abelian $p$-groups

Let $A$ be an Abelian group of prime power order. It can be expressed as a (unique) direct product of cyclic groups of prime power order: $A = \mathbb{Z}_{p^{n_1}} \times \cdots \times ...
1
vote
2answers
74 views

$g\in G$ maximal order in $G$ abelian then $G=\left<g\right>\oplus H$

If $g\in G$ is an element of maximal order in a finite abelian group $G$ then exists $H\leq G$ such that $G=\left<g\right>\oplus H$ Attempt: Using fundamental theorem I know that ...
0
votes
1answer
50 views

Subgroups of finite Abelian groups

I am interested in finding all of the subgroups (up to isomorphism) of a finite Abelian group $A$. I know the following: -- A finite Abelian group $A$ can be represented as a direct product of ...
2
votes
1answer
36 views

Let $G$ be an Abelian group with odd order. Show that $\varphi : G \to G$ such that $\varphi(x)=x^2$ is an automorphism

Let $G$ be an Abelian group with odd order. Show that $\varphi : G \to G$ such that $\varphi(x)=x^2$ is an automorphism. I was able to show that the $\varphi$ function is a homomorphism and ...
10
votes
1answer
107 views

If the tensor power $M^{\otimes n} = 0$, is it possible that $M^{\otimes n-1}$ is nonzero?

Let $M$ be a module over a commutative ring $R$. It is possible that $M \otimes M = 0$ if $M$ is nonzero, for example when $R = \mathbb{Z}$ and $M = \mathbb{Q}/ \mathbb{Z}$. What about when higher ...
1
vote
1answer
23 views

Does this condition gaurantee the cyclicity of a finite abelian group?

Let $G$ be a finite abelian group in which there are at most $n$ solutions of the equation $x^n = e$ for each posivite integer $n$. How to determine if $G$ is cyclic or not?
0
votes
1answer
44 views

Which elements of this cyclic group would generate it?

Let $n$ be a given arbitrary positive integer, and let $U_n$ denote the group of all the positive integers less than $n$ and relatively prime to $n$ under multiplication mod $n$. Then for which values ...
0
votes
2answers
68 views

Possible difference between $\mathbb{Z}$-modules and vector spaces

Suppose $G$ is a free abelian groups, i.e. a free $\mathbb{Z}$-module; we have a set $S \subset G $ such that $S$ spans $G$. Can we conclude that the rank of $G$ as a $\mathbb{Z}$-module is $ \leq ...
0
votes
2answers
38 views

Abelian and residually finite groups

If $G$ is a finitely generated abelian group then $G$ is residually finite. I don't know if the result holds or not. I tried to follow the definition but could not go far. Any hint will be highly ...
2
votes
1answer
49 views

Maximal $\mathbb{Z}$-submodules in $\mathbb{Q}$

Is it true that $\mathbb{Q}$ viewed as $\mathbb{Z}$-module ( i.e. abelian group ) has not maximal $\mathbb{Z}$-submodules ? Why ?
0
votes
1answer
23 views

Order of elements in a disjoint cycle

What's the difference between a subgroup and a cyclic subgroup? $A_4 = \{e,(123),(132),(124),(142),(134),(143),(234),(243),(12)(34),(13)(24),(14)(23)\}$ And if I was looking for a subgroup of order ...
2
votes
4answers
83 views

Order of elements in a disjoint cycle

Just wondering how to find the order of each element in this group: $A_4 = \{e,(123),(132),(124),(142),(134),(143),(234),(243),(12)(34),(13)(24),(14)(23)\}$ I tried writing each elements not in ...
1
vote
1answer
28 views

Let $A$ finitely generated abelian group, and $A_1 \le A$. I have to prove that $rk(A)=rk(A_1)+ rk(A/A_1)$

Let be $A$ a finitely generated (f.g.) abelian group, and let be $T$ its torsion, then by the structure theorem of f.g. abelian gruops we have that $A/T \simeq \mathbb{Z}^d$, so we can define $b$ the ...
4
votes
1answer
39 views

uniqueness of groups in an exact sequence

I was wondering how unique are the groups making up to an exact sequence. Suppose we have three groups $A, B, C$ such that the sequence $$ A \rightarrow B \rightarrow C $$ is exact. I wanted to know ...
6
votes
2answers
58 views

If $G$ is a finite union of some of its abelian subgroups, then the index of the center of the group is finite

If $G$ is a finite union of some of its abelian subgroups, then the index of the center of the group is finite Would I not simply state that by Lagrange's theorem, $Z(G)$ can divide into the ...
3
votes
1answer
38 views

Shorter proof for some equvalences

Let $(G,\cdot)$ be a group show that A) $G$ is abelian B) For all $x,y\in G: (xy)^{-1}=x^{-1}y^{-1}$ C) For all $x,y\in G: (xy)^{2}=x^{2}y^{2}$ D) There existst an $n\in \mathbb{Z}$ such that for ...
3
votes
1answer
108 views

Is the group $(G,*)$ abelian?

Let $(G,*)$ be a finite group in which the sets $C_a$={$x\epsilon G$|$ax=xa$} have the same cardinality, for all $a \epsilon G$ \{e}. My question is: is the group abelian?
1
vote
2answers
74 views

Using the Fundamental Theorem of Finite Abelian Groups

Let $G$ be a finite abelian group and let $p$ be a prime that divides the order of $G$. Use the Fundamental Theorem of Finite Abelian Groups to show that $G$ contains an element of order $p$. The ...
4
votes
2answers
103 views

Explicitly computing the isomorphism class of the tensor product of two finite abelian groups

How do I compute the isomorphism class of $A\otimes_\mathbb{Z} B$, where $A$ and $B$ are abelian of finite order? I can do this for a few examples, but I am unsure of how to proceed in the ...
5
votes
1answer
141 views

Show from the axioms: Addition in a quasifield is abelian

According to wikipedia a quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is a group. (As usual, we denote its identity element by $0$.) $(Q\setminus\{0\},\cdot)$ is a loop. (Its ...
1
vote
1answer
54 views

Do the circle groups have any interesting stand-alone descriptions?

By the circle groups, I mean firstly the circle group $\mathbb{T} \subseteq \mathbb{C}$ of all complex numbers having modulus $1$, and secondly the commutative group $\mathbb{S} = \mathbb{T} \cap ...
0
votes
1answer
61 views

Using the conjugacy class equation [duplicate]

Let $G$ be a group of order $p^2$. Use the class equation to prove that $G$ is abelian. The conjugacy class equation, at least how I remember it, is $$ |G| = |Z(G)|+\sum_{x\in I \backslash Z(g)} ...
3
votes
2answers
52 views

Give an example of a non-abelian group G containing a proper normal subgroup N such that $G/N$ is abelian.

Give an example of a non-abelian group G containing a proper normal subgroup N such that $G/N$ is abelian. I KNOW THERE IS A QUESTION OF THE SAME NAME. However, I need more involved assistance. My ...
0
votes
2answers
58 views

Abstract Algebra. Let $\mathit{G} $ be an abelian group. Show that the elements of finite order in $\mathit{G}$ form a subgroup of $\mathit{G}$.

Let $\mathit{G} $ be an abelian group. Show that the elements of finite order in $\mathit{G}$ form a subgroup of $\mathit{G}$, called the torsion subgroup of $\mathit{G}$. let $g \in G$ I know that ...
2
votes
1answer
100 views

Classifying abelian groups up to isomorphism

List all abelian groups (up to isomorphism) of the given orders: a) $144$, b) $600$ a) For order $144$, I feel confident with this one so far: $\mathbb{Z}_4 \oplus \mathbb{Z}_{36}$ Elementary ...
0
votes
1answer
52 views

Doubt about the tensor product

Suppose $M$ is an abelian group and $F(X)$ is the free abelian group over $X$. Is it true that any element of $M\otimes F(X)$ can be written as a finite sum $$m_{1}\otimes x_{1}+ \cdots+m_{n}\otimes ...
0
votes
1answer
53 views

Show that any subgroup of a finitely generated abelian group is finitely generated?

I am working through Rotman 2.89 and I can't seem to solve this one. Note: Please do not link me to the related questions such as Proving that a subgroup of a finitely generated abelian group is ...
3
votes
1answer
77 views

Finite abelian p-group and an element of maximal order

I'm studying for an exam and I'm having trouble understanding the proof given for the following statement: Suppose $G$ is a finite abelian $p$-group and $a \in G$ has maximum order, then there ...
0
votes
2answers
36 views

Basic proof of statement in abstract algebra?

http://www.proofwiki.org/wiki/Abelian_Quotient_Group The third step (in both proofs) is something I am having trouble seeing. The theorem itself is not difficult to prove, but it is much cleaner this ...
1
vote
1answer
49 views

subgroup proof.

Prove that if $G$ is an abelian group, then $H =\{ x \in G\mid x^{2} = e \}$ is a subgroup of $G$. I did show that $H$ is close, associative, have identity and inverse element. Then my prof said I ...
2
votes
1answer
106 views

Is there a name for the generalization of the concept “Abelian group” where the axiom $-x+x = 0$ is weakened to the following?

Is there a name for the generalization of the concept "Abelian group" where the axiom $−x+x=0$ is replaced by the following list? $−0=0$ $−(x+y)=−x+−y$ $−(−x)=x$ $x+(-x)+x = x$ In multiplicative ...
4
votes
1answer
54 views

Splitting of exact sequence of groups when middle group has split subgroup.

I am trying to show that a short exact sequence of abelian groups splits. I have a short exact sequence, $$0\rightarrow \mathbb{Z} \rightarrow G \rightarrow \mathbb{Z}_2 \rightarrow 0$$ and I know ...
0
votes
1answer
190 views

Question about Finite Abelian Groups [duplicate]

Let $(G, .)$ be a finite abelian group, $G=\{x_1, ..., x_n\}$ and let $x=x_1. \cdots. x_n$. Show that $x^2=e$ Suppose $G$ has no element of order 2 or that $G$ has more than one element of order 2. ...
2
votes
1answer
125 views

Classify Artinian $\Bbb Z$-modules

How can I classify Artinian $\mathbb{Z}$-modules as Noetherian $\mathbb{Z}$-module? (A $\mathbb{Z}$-module is Noetherian iff it is finitely generated). Any hint will be helpful. I have seen the ...
0
votes
3answers
89 views

Commutator Subgroup is Normal Subgroup of Kernel of Homomorphism

Please help to understand this problem. Let $G$ be a group, $H$ an abelian group, $\phi : G \rightarrow H$ a homomorphism. Show that $C(G) \lhd \mathrm{Ker}(\phi)$ I must be misunderstanding ...
0
votes
2answers
56 views

“Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$”

I have a question that says this: Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$ according to the fundamental theorem of finitely generated abelian groups. ...
0
votes
1answer
52 views

If a group has only one commutator, why does that mean it is abelian?

I understand that if $aba^{-1}b^{-1} = e$ then $ab$ is commutative, but I don't see how having multiple commutators will prevent the group from being abelian
3
votes
1answer
96 views

an order of automorphism group of finite abelian group

This is problem of Rotman's Exercise 7.9(i). If $G$ is finite abelian group with $|G| >2$, then $\operatorname{Aut}(G)$ has even order. How can I approach to this problem? Could you suggest ...
1
vote
0answers
56 views

Working with special cases of the converse of Lagrange's theorem

I am to answer true/false to statements on the form: Every abelian group of order divisible by $n$ contains a cyclic subgroup of order $n$. This follows directly from Cauchy's theorem when $n$ ...
2
votes
2answers
81 views

The sum of the orders of all elements of a group G

Let $Z$ be a finite group and denote $k(Z)$ the sum of the orders of all the elements of the group $Z$. I have to determine min $k(Z)$ and max $k(Z)$ when $G$ goes through the set of the abelians ...