Should be used with the (group-theory) tag. A group $(G,*)$ is said to be abelian if $a*b=b*a$ for all $a,b\in G.$

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1answer
92 views

Abelian group and number of solutions of $x^n=e$

Let $G$ be an abelian group in which the number of solutions of $x^n=e$ is at most $n$ for every positive integer $n$. Show that $G$ is cyclic. I've shown this nice fact only when $|G|$ is finite. ...
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2answers
54 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
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1answer
171 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 ...
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2answers
146 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 ...
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2answers
38 views

Ordered abelian groups

Consider the following axioms: 1) $\ x+(y+z)=(x+y)+z$ ; $\forall x \forall y \forall z$ 2) $\ x+0=x$ ; $\forall x$ 3) $\forall x$ $ \exists y$ such that $\ x+y=0$ 4) $ \ x+y=y+x$ ...
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5answers
190 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. $without$ $theory!$
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1answer
57 views

Two free basis of a free abelian group

We have a free abelian group $A(X)$, where $X$ is its free basis, and let $Y$ another free basis for $A(X)$. We know that every $g\in A(X)$ can be expressed as $g=a_1x_1+...+a_nx_n$ where the $a_i$'s ...
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2answers
237 views

Unsatisfying proof on project crazy project (finite group is not divisible)

I found this proof that no finite group is divisible.(here) Let A be a finite divisible group then there are elements $x_k$ such that $x_k^k=1$ for each natural k. Then that would mean there is an ...
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2answers
78 views

Abstract algebra question: abelian group.

$H=\left\{\begin{pmatrix}1 & b \\ 0 & 1\end{pmatrix} : b \in\mathbb{R}\right\}$ $G=\left\{\begin{pmatrix}a & b \\ 0 & d\end{pmatrix}: a, b, d \in\mathbb{R}, ad\ne0\right\}$ $H$ is ...
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4answers
410 views

Why is $\operatorname{Hom}(A, B)$ an abelian group?

Can someone please explain why a Hom-set (the set of all morphisms between two abelian groups $A$ and $B$) does also form an abelian group with addition? By the way both groups $A$ and $B$ have the ...
3
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1answer
155 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 ...
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1answer
81 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 ...
1
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1answer
75 views

Proof of Abelian group

I have run into something I don't understand - a proof that group $P= \{a+b\sqrt5: a,b \in \Bbb Q\}$ is abelian considering usual addition operator $+$. Authors state that checking if the difference ...
2
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1answer
630 views

Prove that if a group is nilpotent , then its quotient with its Frattini subgroup is abelian

I know that : 1) Nilpotent group is solvable. 2) Subgroup of a solvable group is solvable. 3) Solvable and simple group is abelian. Now I should use these facts to prove it.
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2answers
583 views

Prove that the only homomorphism between a simple non-abelian group G and abelian group A is trivial

Prove that the only homomorphism between a simple non-abelian group $G$ and abelian group $A$ is trivial. OK. So G is a perfect group (G' = G) and A is abelian (A' = {1})
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4answers
255 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
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1answer
127 views

Find a torsion free, non cyclic, abelian group $A$ such that $\operatorname{Aut}(A)$ has order 2

Is there any chance to find a torsion free, non cyclic, abelian group $A$ such that $\operatorname{Aut}(A)=\mathbb Z_2$? ($\mathbb Z_2$ is the cyclic group of order $2$) Notation ...
2
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2answers
93 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 ...
2
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3answers
76 views

If $ \langle G, \star \rangle $ is an abelian group, then for all $a, b \in G$, show that $(a \star b)^{n} = a^{n} \star b^{n}$.

If $ \langle G, \star \rangle $ is an abelian group, then for all $a, b \in G$, show that $(a \star b)^{n} = a^{n} \star b^{n}$. I am stuck at the first step, unable to figure out how to start. I am ...
4
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3answers
176 views

Examples of loops which have two-sided inverses.

Are there any neat examples of non-associative loops such that for each element a in the loop there exists $a^{-1}$ so that $a*a^{-1}=1=a^{-1}*a$. Even cooler would be a commutative loop. Also: are ...
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0answers
55 views

Question about finitely generated abelian groups

Why is an Abelian group finitely generated iff $A/mA$ is finite for some $m\gt 1$ and $A$ has a norm function? I know that $mx$ where $x$ is an element of $A$ is equivalent to $0$ in $A/mA$, and I ...
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1answer
81 views

If $f\in\hbox{Hom}_{\mathbb{Z}}(\prod_{i=1}^{\infty }\mathbb{Z},\mathbb{Z})$ and $f\mid_{\bigoplus_{i=1}^{\infty } \mathbb{Z}}=0$ then $f=0$.

Prove that if $f\in \hbox{Hom}_{\mathbb{Z}}(\prod_{i=1}^{\infty }\mathbb{Z},\mathbb{Z})$ and $f\mid_{\bigoplus_{i=1}^{\infty } \mathbb{Z}}=0$ then $f=0$. I took an element of $\prod_{i=1}^{\infty ...
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1answer
117 views

Torsion subgroups of finitely generated abelian groups

What information one can get about the torsion subgroups from a short exact sequence of finitely generated abelian groups ?
3
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1answer
95 views

Fibered coproducts in $\mathsf{Ab}$

Aluffi (II.3.9) asks Show that fiber products and coproducts exist in $\mathsf{Ab}$. (Cf. [exercise on fiber products and coproducts in $\mathsf{Set}$]). The equalizers for $\mathsf{Ab}$ and ...
4
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1answer
44 views

Can this be proved purely on base of UMP?

Let $A,B$ be abelian groups and let $P$ serve as a product with projections $p_{A}:P\rightarrow A$ and $p_{B}:P\rightarrow B$. Let $C$ be an abelian group and let $f:C\rightarrow A$ and ...
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2answers
104 views

Is a set that is an abelian group under addition and a group under multipliation a field?

I suspect the answer to my question is yes, but I'm just checking my understanding. If we have a set which is an Abelian group under addition and a group under multiplication is it then defined as a ...
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1answer
144 views

Prove a group is an abelian group

Let $G\subseteq \mathbb N.$ How do I prove that $G$ is an abelian group with respect to the binary operation " * " defined by $\;a*b = a+b+11$ ?
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3answers
112 views

Are $\Bbb Z_{8} \times \Bbb Z_{10} \times \Bbb Z_{24}$ and $\Bbb Z_{4} \times \Bbb Z_{12} \times \Bbb Z_{40}$ isomorphic? [closed]

Are the groups $\Bbb Z_{8} \times \Bbb Z_{10} \times \Bbb Z_{24}$ and $\Bbb Z_{4} \times \Bbb Z_{12} \times \Bbb Z_{40}$ isomorphic? Why or why not? (Here $\times$ means the direct product or direct ...
4
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1answer
69 views

Computation of a homology group of a simple complex $0\rightarrow \mathbb{Z}^l \rightarrow \mathbb{Z}^n \rightarrow \mathbb{Z}^m\rightarrow 0$

Consider the following sequence of abelian groups, where $f\circ g = 0$. $$0\longrightarrow \mathbb{Z}^l ...
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0answers
191 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 ...
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1answer
402 views

Prove that is an abelian group

Given the binary operation in $a \circ b \in \mathbb{R}$ defined by $a \circ b = a + b + \pi$, prove that with this operation in $\mathbb{R}$, it's an abelian group. I know that: Given a group $G$ ...
4
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2answers
114 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?
4
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1answer
259 views

Isomorphic finite abelian groups [duplicate]

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, ...
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1answer
90 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 ...
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3answers
244 views

show that rational numbers with the multipiciation are not abelian finitely generated group

we need to show that $( Q \ast , \bullet )$ is not abelian finitely generated group for all finite subset $S \subseteq Q \ast $ $ Q\ast=Q \setminus \big\{0\big\} $
2
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2answers
123 views

Let $A$ be a finitely generated abelian group. Show that $\operatorname{Hom}(A,Z)$ 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 ...
1
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1answer
100 views

Let $A$ be an abelian group. Show that $\mathrm{Hom}(\mathbb Z, A)$ is isomorphic to $A$.

Let $A$ be an abelian group. Show that $\mathrm{Hom}(\mathbb Z, A)$ is isomorphic to $A$. My problem is figuring out how to define Φ and using it show the homomorphism between ...
2
votes
0answers
117 views

Epimorphism (abelian group)

Let $(G,\cdot), (H,*)$ Groups and $f: G\rightarrow H$ an Epimorphism. Show that: If G is an abelian group, then H is also an abelian group. Is the reversal of this proposition also true? My idea: ...
4
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2answers
510 views

Tensor product of a finitely generated abelian group and the field of rational numbers

Let $G$ be a a finitely generated abelian group. Then $G\otimes_\mathbb{Z} \mathbb{Q} = 0$ if and only if $G$ is a finite group. The "if" part is easy. The "only if" part can be proved using the ...
5
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3answers
366 views

An Example of Abelian Group with exactly one maximal subgroup.

Let $G$ be a finitely generated group and $G$ has exactly one maximal subgroup. Then I can conclude that $G\cong\mathbb Z_{p^k}$. Now, I am looking for an example of infinite abelian group $G$ such ...
6
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1answer
227 views

Let G be a group of order $n$, where $n$ is a positive integer relatively prime to $\varphi(n)$. Show that G is cyclic.

Let G be a group of order $n$, where $n$ is a positive integer relatively prime to $\varphi(n)$. Show that G is cyclic. You may only assume the Feit-Thompson theorem here and prove in the following ...
3
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1answer
155 views

Group with $a=a^{-1}$ for all $a\in G$ is abelian

Let $G$ be a group, and suppose that $a= {a^{-1}}$ for every $a\in G$. Prove that $G$ is Abelian. I know that I need to prove that ${a^{-1}}$ = a and use right multiplication. I also know that I ...
2
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1answer
168 views

What motivates the study of Abelian groups?

Monoids arise naturally as endomorphism monoids, and groups arise naturally as automorphism groups. These are among the primary motivators for their study, in my opinion. What are the (main) ...
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2answers
161 views

The number of elements of a finite group which is a quotient of a finitely generated free abelian group

Let $G$ be a finitely generated free abelian group. Let $\omega_1,\cdots,\omega_n$ be its basis. Let $\alpha_1,\cdots,\alpha_m$ be a finite sequence of elements of $G$. Suppose $\alpha_i = \sum_j ...
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6answers
2k views

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

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 ...
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3answers
95 views

Finding a non-Abelian group

I am stuck at another problem in my homework. Find a non-Abelian group of size 48 such that the order of its elements are either 1, 2, 3 or 6. I need some hints/tips to start on this problem as ...
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3answers
80 views

Finding an Abelian subgroup

I am working on my homework and am currently stuck at the following question: Let $\mathcal S_{10}$ be the group of all permutations of the set of elements {1, 2, ..., 10}. Find an Abelian ...
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2answers
132 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 ...
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3answers
359 views

Find all possible abelian groups of order $120$.

Find all possible abelian groups of order $120$. If someone could walk me through how to do this, that would be great.
1
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2answers
67 views

Properties of Order of a Group

Having real trouble getting started on this question, even though it doesn't seem hard: Let $g,h \in G$ where $G$ is an Abelian group. Then assume that $ord(g), ord(h)$ are finite with ...