2
votes
1answer
28 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 * ...
0
votes
2answers
41 views

If $G$ is non-abelian, then $Inn(G)$ is not a normal subgroup of the group of all bijective mappings $G \to G$

Let $(G,\cdot)$ be a group and let $\mathfrak{S}(G)$ be the set of all bijective mappings from $G$ to $G$. Show that: If $G$ is non-abelian, then $Inn(G):=\{\kappa_a \vert a\in G\}$ is not a ...
1
vote
1answer
33 views

Number of elements in a group

The group $G$ consists of the binary strings of length $5$ under addition $\mod 2$ in each component. (It is isomorphic to $(\mathbb Z_2)^5$, the direct product of $5$ copies of $\mathbb Z_2$.) Let ...
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 ...
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 ...
2
votes
1answer
37 views

Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. Prove that $G $ is an abelian group.

Let G be a group, where $(ab)^3=(a^3)(b^3)$ and $(ab)^5=(a^5)(b^5)$. Prove that $G$ is abelian group. Thank you in advance. Any help is appreciated.
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 ...
0
votes
2answers
64 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
37 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
48 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
38 views

Commutative group

Let $A$ be an nonempty set, and $f: A^3 \rightarrow A$ mapping which satisfies: $f(x,y,y)=f(y,y,x)$ for each $x,y \in A$ ...
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
78 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
24 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
38 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
57 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
29 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?
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 ...
3
votes
1answer
50 views

isomorphism between divisible, totally ordered, abelian groups

Let $G$, $H$ be divisible, abelian, linearly ordered groups, whose cardinalities are equal and satisfy $\mu := |G|=|H|>\aleph_{0}$. These are supposed to be (order!) isomorphic. And just about ...
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.
0
votes
2answers
55 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
96 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 ...
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 ...
-2
votes
1answer
36 views

Prove that the following are equivalen for abelian group

Let $(G, *)$ be a group. Prove that the following are equivalent: a. $G$ is abelian. b. $aba'b' = e$ for all $a,b \in G$. c. $(ab)^{2}$ = $a^2b^2$ for all $a, b \in G$.
1
vote
1answer
48 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 ...
4
votes
1answer
52 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
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. ...
2
votes
1answer
123 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
84 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 ...
1
vote
1answer
64 views

Prove that a group is abelian if every element commutes with exactly K other elements

Let $(G,*)$ be a finite group with the property that every element, aside the neutral element $e$, commutes with exactly K other elements. Alternatively speaking, the centralizer of every element is a ...
0
votes
2answers
53 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
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 $n$ ...
1
vote
0answers
46 views

Find which of the abelian groups are isomorphic to $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}_{6},\mathbb{Q}\oplus \mathbb{Z}_{3})$

Which of the abelian groups are isomorphic to $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}_{6},\mathbb{Q}\oplus \mathbb{Z}_{3})$?
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 ...
6
votes
2answers
49 views

Divisible groups, exercise from Rotman's theory of groups

The following exercise is from Rotman, An Introduction to the theory of groups, 4th ed, p324. "The following conditions on a group G are equivalent: (i) G is divisible, (ii) Every nonzero quotient of ...
5
votes
1answer
53 views

Why is a normal subgroup of $G_1\times G_2$ with trivial intersections with $G_1$ and $G_2$ is abelian?

Let $G=G_1\times G_2$ be a direct product, and let $H\triangleleft G$ be a normal subgroup such that $H\cap G_1=H\cap G_2=\{1\}.$ Then $H$ is abelian. I considered the commutators of two ...
1
vote
1answer
75 views

Non-abelian group in which $\forall_{a,b\in G} (ab)^3=a^3b^3$ [duplicate]

Give an example of a non-abelian group, in which $(ab)^3=a^3b^3$ for every element $a,b$ in $G$. I understand that such a group should be of order divisible by 3 (see Problem from Herstein on group ...
1
vote
0answers
105 views

Isomorphism theorem for Abelian groups, related to Hatcher exercise 2.1.14

I am trying to understand which Abelian groups can fit the short exact sequence \begin{equation} 0 \rightarrow \mathbb{Z}_{p ^m}\rightarrow A \rightarrow \mathbb{Z}_{p^n}\rightarrow 0. \end{equation} ...
3
votes
3answers
85 views

Isomorphisms of direct products of finite abelian groups

Suppose $G_1, G_2, H_1, H_2$ are finite abelian groups with $G_1 \times G_2 \cong H_1 \times H_2$, and $G_1 \cong H_1$. Prove that $G_2 \cong H_2$. Since the groups are finite, the isomorphisms ...
0
votes
1answer
57 views

Show that $G$ has exactly one subgroup of order $8$.

I have this problem: Let $G$ be an abelian group of order $72$. Show that $G$ has exactly one subgroup of order $8$. I've seen how to find all abelian groups (up to isomorphism) of order $n$, ...
1
vote
1answer
71 views

Subgroup of an abelian Group

I think I have the proof correct, but my group theory is not that strong yet. If there is anything I am missing I would appreciate you pointing it out. Let $G$ be an abelian group (s.t. $gh = hg$ ...
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 ...
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 ...