2
votes
2answers
76 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
117 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
64 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
47 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 ...
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
42 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 ...
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
73 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
99 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 ...
0
votes
1answer
56 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)} ...
0
votes
1answer
30 views

Subgroups of a finite abelian group

Let $G$ be a finite abelian group, and let $K$ be a subgroup of $G$. Does $G$ necessarily have a subgroup $H$ such that $H\cong G/K$ and $H\cap K=\langle 0\rangle$? I'm not sure where to start.
3
votes
1answer
73 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
1answer
189 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. ...
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
78 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 ...
1
vote
2answers
87 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 ...
0
votes
0answers
20 views

subgroup of character group

Let $B$ be a subgroup of a finite Abelian group $A$ and $$B^0=\{\alpha \in \hat{A}\mid \alpha(b)=1 \text{ as } b \in B\}$$ Prove a. $B^0$ is subgroup of $\hat{A}.$ And for any subgroup ...
1
vote
1answer
37 views

Under what conditions should a sub-group of a direct sum, itself be a direct sum?

This is a question I'm struggling a couple of days with: Let $G_1,G_2$ be abelian groups, and let $H$ be a subgroup of $G:=G_1\oplus G_2$. Under what conditions must $H$ be a group of the form ...
5
votes
1answer
75 views

Prove a result on the size of the minimal set that generates a finite abelian group

I am asked to prove the following: Let $G$ be a non-trivial, finite abelian group. Let $s$ be the smallest positive integer such that $G = \langle a_1,...,a_s\rangle$ for some $a_1,...,a_s \in G$. ...
2
votes
2answers
66 views

When is the automorphism group $\text{Aut }G$ cyclic?

Let $G$ be a finite group. Under which conditions on $G$ is the automorphism group $\text{Aut }G$ cyclic? More precisely, does $G$ is abelian or $G$ is cyclic implies $\text{Aut }G$ is cyclic?
-4
votes
1answer
70 views

Is this group: $\mathbb Z^{*}_{15}$, is cyclic? [closed]

Is this group: $\mathbb Z^{*}_{15}$, is cyclic? I tried to find a generator. and i didn't found one. But how can i prove that this group is not cyclic?
0
votes
2answers
31 views

In the group A/B find the order of the coset (x$_1$+2x$_3$)+B

A is an abelian free group, with the base x$_1$,x$_2$,x$_3$. will be B a sub-group that created with x$_1$+x$_2$+4x$_3$,2x$_1$-x$_2$+2x$_3$. In the group A/B find the order of the coset ...
4
votes
1answer
97 views

If $σ^2$ is the identity map from $G$ to $G$, prove that $G$ is abelian.

Let $G$ be a finite group which possesses an automorphism $σ$ such that $σ(g)=g$ if and only if $g=1$. If $σ^2$ is the identity map from $G$ to $G$, prove that $G$ is abelian. This is what I got ...
0
votes
3answers
113 views

Show that $G\{m\}$ is finite for all non-zero $m\in \mathbb{Z}$

Let $G$ be an abelian group, and let $m$ be an integer, then we define $G\{m\} := \{a\in G:ma=0_G\}$. Now, suppose that $G$ is an abelian group that satisfies the following properties: (i) For all ...
1
vote
5answers
102 views

can somebody recommend a book in a group theory.

can somebody recommend a book in a group theory. that include just questions and their answers. $without$ $theory!$
3
votes
1answer
57 views

Nilpotent action on $p$-group

Let $A$ be a finite, abelian $p$-group and $\Gamma$ is a multiplicative topological group isomorphic with the additive group of $p$−adic integers $\mathbb Z_p.$ and let $\gamma_0$ a topological ...
0
votes
1answer
56 views

If $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$.

Let $G$ be finite abelian group and $\hat G$ be its character group. I need hint proving that if $a\in G$ and $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$ (the identity element). I can prove it ...
-3
votes
3answers
88 views

Let $G = \{1, a, b, c\}$ be a group of order 4…Exist two groups of order $4$.

Let $ G = \{1, a, b, c\}$ be a group of order 4. Show that, if $G$ is cyclic $G \cong \mathbb Z_4,$ and if $G$ is not cyclic then $G \cong K_4.$ It now follows that there are only 2 groups of order ...
7
votes
4answers
168 views

Let $G$ be a finite group which has a total of no more than five subgroups. Prove that $G$ is abelian.

Let $G$ be a finite group which has a total of no more than five subgroups. Prove that $G$ is abelian. I can prove that if $\left|G\right|\leq5 $ then $G$ is abelian. Is it equivalent to this ...
2
votes
2answers
82 views

Finite abelian groups of odd order

I am reading this paper. It is about finite abelian groups of odd order. I need to find maximal subset which doesn't contain 3-term arithmetic progression. I don't understand the need of odd order. I ...
4
votes
0answers
101 views

$(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ is a cyclic group

I would like to prove that the group $(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ of the invertible elements of $\mathbb{Z}/p^r\mathbb{Z}$ with $p>2$ prime and $r>0$ is cyclic. My text suggests to ...
4
votes
2answers
89 views

Commutativity of a finite group

In a finite group a representative can be chosen from each conjugacy class such that they all commutate. Prove that the group is commutative. Does this still hold true if the group is infinite?
2
votes
1answer
92 views

Isomorphic finite abelian groups

Let $G$ and $H$ be finite abelian groups. Show that if for any natural number $n$ the groups $G$ and $H$ have the same number of elements of order $n$, then $G$ and $H$ are isomorphic. I know, ...
1
vote
1answer
52 views

Structure of $C(F_5)$ from Rational Points on Elliptic Curves

In the book Rational Points on Elliptic Curves by Silverman/Tate one examines the elliptic curve $y^2 = x^3 + x + 1$ over $F_5$. One can then easily determine the group $$ C(F_5) = \lbrace ...
1
vote
2answers
60 views

Let A be a finitely generated abelian group. Show that Hom(A,Z) is a free abelian group.

My question is Let $A$ be a finitely generated abelian group. The structure theorem says that $A$ is isomorphic to $F \times T$, where $F$ is isomorphic $\mathbb Z^m$, some $m \geq 0$, and $T $ is ...
2
votes
2answers
337 views

Prove that a finite abelian group is simple if and only if its order is prime.

So I'm having trouble with this problem. I know that the definition of a simple group means that the group has no nontrivial subgroups. I know that this can be proven somehow with the help of the ...
0
votes
0answers
20 views

Representatives of conjugacy class commute [duplicate]

Let $g_1,g_2,\ldots,g_r$ be representatives of the conjugacy classes of the finite group $G$ and assume that these elements pairwise commute. Prove that $G$ is abelian.
1
vote
2answers
39 views

multiplicative order in field

Let $\alpha$ be primitive element of GF(7). Then order of $\alpha$ is 6, i.e. $o(\alpha)=6$. Now we know that $\alpha^4$ is not equal to 1, and that $o(\alpha^4)$ = $\frac{6}{gcd(6,4)}$. This also ...
1
vote
0answers
67 views

Orders of Elements in Minimal Generating sets of Abelian p-Groups

I'm looking for as much information about the orders of elements in minimal generating sets of finite abelian $p$-groups as possible. What I really need is complete knowledge about the possible orders ...
1
vote
1answer
62 views

Cyclic group of order $6$ defined by generators $a^2=b^3=a^{-1}b^{-1}ab=e$.

The cyclic group of order $6$ is the group defined by the generators $a,b$ and relations $a^2=b^3=a^{-1}b^{-1}ab=e.$ For this problem I am assuming that it must be $a,b \neq e$. I will show the ...
3
votes
2answers
72 views

Order of a product in an abelian group.

Suppose $G$ is a finite abelian group and has two element $a$ and $b$, such that $\circ(a)=m$ and $\circ(b)=n$ and $lcm(m,n)\neq m,n$. Is it true that $\circ(ab)=lcm(m,n)$? Thanks in advance.
2
votes
2answers
80 views

Fundamental Theorem of Finite Abelian Groups

Fundamental Theorem of Finite Abelian Groups indicates that $\mathbb{Z}_{n}$ is isomorphic to $\mathbb{Z}_{p_1^{k_1}} \times \mathbb{Z}_{p_2^{k_2}}\times$ ... $\times\mathbb{Z}_{p_n^{k_n}}$ where ...
0
votes
2answers
70 views

Question about Abelian group proof

I prove that if $G$ is Abelian group so if $a,b\in G$ has a finite order so $ab$ has a finite order to.. (Maybe later I'll upload here my proof to see of she is correct....) Now, I have to show that ...
2
votes
0answers
48 views

Characters of subgroups of finite abelian groups

Let $G$ be a finite abelian group. Let $H$ be a subgroup of $G$. Let $\hat{G}$ be the group of characters of $G$. Is there a character $\chi \in \hat{G}$ such that $\chi(g) = 1$ iff $g \in H$?
0
votes
0answers
67 views

Showing that $ U(2^n) $ is not a cyclic group for $ n \geq 3 $ [duplicate]

Could anyone please explain to me why $ U(2^n) $ is not a cyclic group for $ n \geq 3 $? I need help on this because I have an algebra exam tomorrow. Thanks!
2
votes
1answer
318 views

Subgroups and quotient groups of finite abelian $p$-groups

Motivation The fundamental theorem of finite abelian groups gives us a concise description of the isomorphism types of finite abelian $p$-groups $G$ (in the following, $p$ is a fixed prime). The ...
1
vote
0answers
70 views

Structure of finite abelian group

I am trying to solve this problem, and I think I should use the structure theorem for finite abelian groups, but i can't really figure this out. Let $G$ be a finite abelian group. Prove there is a ...
0
votes
0answers
71 views

Exact Sequence of Finite Abelian Groups

I was reading a paper and in it the author arrives at an exact sequence of abelian groups all with orders dividing the same prime $p$. For convenience, I will write the groups as a tuple ...