Questions tagged [abelian-groups]

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

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Pure Mathematics questions on Group Theory

I am learning Group Theory but am stuck on this particular practice questions, however I am stuck on a few of these. For the second part I have proved that theta is bijective and that theta(x ** y)=...
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Isomorphism between $SO_2\tilde{\times}\mathbb{Z}_2$ and $O_2$

This is the exercise 23.10 p. 135 of Groups and symmetry of Armstrong : Let $G$ be an abelian group and write $G \tilde{\times}\mathbb{Z}_2 $ for the semidirect product $G\rtimes_\phi\mathbb{Z}_2$, ...
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28 views

Product $PN$ of normal subgroups is abelian

I am trying to show that every non-abelian group $G$ of order $6$ has a non-normal subgroup of order $2$ using Sylow theory. First, Sylow's Theorem says the number of Sylow $2$-subgroups $n_2$ is ...
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Prove: If gcd(ord(x),ord(y))=1, then ord(xy)=ord(x) ord(y) when x,y in G (abelian) [duplicate]

$\newcommand{\ord}{\operatorname{ord}}$Let G be an abelian group and let $x,y \in G$ be two elements with finite order. Then, prove that if $\gcd(\ord(x),\ord(y))=1$, then $\ord(xy)=\ord(x) \ord(y)$ ...
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Grothendieck group “commutes” with direct sum

The Grothendieck completion group of a commutative monoid $M$ is the unique (up to isomorphism) pair $\langle \mathcal{G}(M), i_M\rangle$, where $\mathcal{G}(M)$ is an abelian group and $i_M\colon M\...
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How to show that $H=\langle a,b\rangle=\{a^m\cdot{b^n}\mid m,n\in{\mathbb{Z}}\}$ is a subgroup of $\langle G,\cdot\rangle$?

I have $a,b\in{G}$ and also $G$ is Abelian. I thought that first I should show that $H\subset{G}$ and then show that $\forall{h_1,h_2}\in{H}$ we will have $ h_1\cdot{h_2}\in{H}$. First to show that $...
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What does the abelianization mean?

Abelianise each of: (a) $\Bbb Q \times S_4$ (b) $D_{12} \times A_4$ (c) $G \times Z_{10}$, where $G$ is the dicyclic group of order 12 and write down the torsion coefficients of the resulting ...
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Order of a subset

Hihi, im in a trouble with a excercise. I hope someone can help me: Let (G, $\bullet$) be a finite group, and U a subset of G. $\\$ Let $\phi$ : G $\times$ U $\to$ U be a group action such that $\...
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How to construct additive inverse in an abelian category

A (locally small) abelian category has a canonical addition structure on its hom sets. In fact, this structure forms an abelian group. Let $A$ be an abelian category. Here, an abelian category is a ...
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Is a group abelian? [closed]

Let $G$ be a non-abelian group. Show that at most $\frac{1}{4}$ of the elements of $G$ can be in $Z(G)$. Give an example of a group where exactly $\frac{1}{4}$ of its elements are in $Z(G)$.
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Representing a non-abelian group as a free group [closed]

Can we express the group $(\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes \mathbb{Z}_q$, where $p,q$ are odd distinct primes, using a free group? Thanks a lot in advance.
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Inverse of modulo sum of elements from a cyclical multiplicative group without descrete log

So the question: Given a hash function $$h(x) = \sum_{i=1}^n g^{x_i} \mod p$$ where p is a prime number and g is a generator of the multiplicative cyclic group $\mathbb{Z}_p^×$. n is the length of the ...
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$RHom(A,\mathbb{Z})=0$ implies $A=0$? [duplicate]

Let $A$ be a countably generated abelian group, and both $Hom(A,\mathbb{Z})$ and $Ext(A,\mathbb{Z})$ vanish. Is it true that $A$ is trivial?
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If $G$ is an Abelian group and contains cyclic subgroups of order 4 and 6, what other sizes of cyclic group G must contain?

Let $G$ be an abelian group. Let $a,b \in G$ such that $|a|=4, |b|=6$. $(ab)^{24}=a^{24}b^{24}=e$, identity of G. Hence $|ab|$ can be $1,2,3,4,6,8,12,24$ If $|ab|=8$ then $(ab)^8=e\implies e=a^8b^8=...
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Is there a name for this kind of subgroup?

Let $G$ be an abelian group, $H\subset G$ a subgroup such that if $nx\in H$ for $n\in \bf{Z}$ and $x\in G$ then $x\in H$. Is there a name for subgroups of abelian groups $H$ with this property?
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Describe the types of groups with 10 elements $ (G,•)$ group and $|G|=10$. [closed]

I wrote the divisors of $D_{10}= 1,2,5,10$, but I don't know how I can continue.
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Let G be a Group and $a,b \in G$. If $a$ and $b$ commute and $|a|$ and $|b|$ are finite. What can be possible values for $|ab|$? [duplicate]

I want to find the order of $ab$, which I have tried to find as follows. Let order of $a$ be $|a|=m$ and that of $b=|b|=n$ Let $x=\text{lcm} (m,n)$, then clearly $\exists$ integers $ s , t $ such ...
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Algebra topic suggestions- Final project [closed]

Good day. My abstract algebra teacher asked me to do a mini project, I'm looking for a topic that covers all (or most) of the following topics: Free modules. Matrix bases and finitely generated ...
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If $G$ is an Abelian Group of rank $r$ then $G\otimes_\mathbb{Z}\mathbb{Q}$ is isomorphic to $\mathbb{Q}^r$

So I'm trying to prove that if $G$ is an Abelian Group of rank $r$ (As $\mathbb{Z}$-module) then $G\otimes_\mathbb{Z}\mathbb{Q}$ is isomorphic to $\mathbb{Q}^r.$ Using the results I know about ...
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1answer
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G is a commutative group of order 2*N, how to prove G has a N-order quotient group? [closed]

$G$ is a commutative group of order $2N$, how to prove $G$ has a $N$-order quotient group? Now I know $G$ must has two subgroup, one is order $N$ and another is order $2$; now what should do to prove ...
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Let $G$ be a non-nilpotent group where all the non-normal abelian subgroups of $G$ are cyclic. Then $G$ has cyclic center.

Theorem : Let $G$ be a non-nilpotent group such that all the non-normal abelian subgroups of $G$ are cyclic. Then $G$ has cyclic center. Proof. Suppose that $Z(G)$ is non-cyclic. since $G$ is non-...
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The group of non zero real or imaginary numbers

I have recently come across with the multiplicative group formed by the union of the real and imaginary axes minus the origin of the complex plane. Surprisingly (at least to me) I couldn't find any ...
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1answer
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Show $G\cong \ker(f) \times \mathbb{Z}$ for abelian $G$

First of all, I am aware of the First Isomorphism Theorem but I am not sure how to use it/if it is useful here $G$ is an abelian group and $f:G\rightarrow\mathbb{Z}$ is a surjective group ...
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Homology with Coefficients of an abelian group

I am trying to see why this is true, this is done in Brown's Cohomology of groups: (In terms of notation $\tau(y)$ is the divided polinomyal algebra). Consider the study of $H_*(G,\mathbb{Z}_p)$, ...
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If $N$ is a normal subgroup of $G$, and $N \cap [G,G]=\{e\}$, then $N$ is contained in $Z(G)$.

I found an answer to this here: If the intersection of a normal subgroup and the derived group is $\{e\}$, show that $N$ is a subset of $Z(G)$.. However I don't really understand some of the answers ...
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A $p$-group of exponent $p$

I saw a theorem For odd $p$, a $p$-group possesses a characteristic subgroup $D$ of class at most $2$ and of exponent $p$ such that every nontrivial $p’$-automorphism of $G$ induces a nontrivial ...
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Short exact sequences of finite abelian groups

I was reading this post, and I was wondering if you had instead the direct sums as such: $$0\rightarrow\mathbb{Z}_{p^{a_1}}\oplus...\oplus\mathbb{Z}_{p^{a_n}}\rightarrow\mathbb{Z}_{p^{b_1}}\oplus...\...
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Applying the subgroup test with an Abelian group [duplicate]

My question states, let $A$ be an Abelian group and $T=\{a\in A | a^5=e\}$, show that $T$ is a subgroup of $A$. I know for the subgroup test I have to prove the identity, that $T$ is closed under $*$,...
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2answers
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Finding the group order from the presentation of a group and deciding if it is abelian

I'm trying to understand more about the presentation of a group. I understand most of the examples that I have seen but I have come across one that I don't understand. Any help would be greatly ...
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1answer
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Clarification on proof of fundamental theorem of finite abelian groups

Herstein's Topics in Algebra provides a proof of the fundamental theorem of finite abelian groups, that is, every finite abelian group is the direct product of cyclic groups. In an earlier exercise, ...
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What's the form of the subgroups of $(\mathbb{Q}_p/\mathbb{Z}_p)^2$?

Let $p$ be prime integer. I've read that every subgroup of $(\mathbb{Q}_p/\mathbb{Z}_p)^2$ has the form \begin{equation} (\mathbb{Q}_p/\mathbb{Z}_p)^e\times U \end{equation} with $0\le e \le 2$ and $...
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On the proof of classification of finitely generated abelian groups

My question: How $K = d_1 \mathbb{Z} \times d_2 \mathbb{Z} \times \dots d_s \mathbb{Z}$ especially when ${\{x_1,x_2, \dots, x_n}\}$ is not a basis that we choose arbitrarilry, i.e. it is such a basis ...
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1answer
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Group of order $pq, p \leq q, p \not\mid q - 1$ is abelian

This is an example from D&F before Cauchy's theorem and before the Sylow theorems. They first note that $Z(G) = \{e\}$, as otherwise $G/Z(G)$ is cyclic and therefore abelian. Sure. They then say ...
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Let $f:G \to H$ where $|G|=pq$ where $p,q$ primes, then $H \cong G$ or $H$ abelian.

Let $f:G \to H$ be a morphism of groups where $f$ is surjective and $|G|=pq$ where $p$ and $q$ are primes, then $H$ is abelian or $H \cong G$. Regard this is a well studied group I have never deal ...
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The longest cycle and relation between the generating elements of a Cayley graph

Let $G$ be a finite group and $S$ be a subset of $G$. We define the Cayley graph of $G$ with respect to $S$ as follows, provided that $1 {\not\in} S$ and $S$ is inverse closed. Definition: The Cayley ...
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1answer
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Number of group homomorphism from $ \mathbb{Z}_{2} * \mathbb{Z}_{2} \to D_{8}$.

How many group homomorphism are there from $ \mathbb{Z}_{2} * \mathbb{Z}_{2} \to D_{8}$?. There are $5$ elements of order $2$ in $D_{8}$ and any non-trivial element in $\mathbb{Z}_{2} * \mathbb{Z}_{...
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Proof that a finite abelian group has a subgroup of order $n$ for every $n$ which divides $|G|$ [duplicate]

As per the title, I'm trying to prove that a finite abelian group has a subgroup of order $n$ for every $n$ which divides $|G|$, and I think I have a solution, but I would like someone to verify this ...
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Explicit description of Barr's *-autonomous category of topological groups

In his paper On duality of topological groups, Micheal Barr describes a certain *-autonomous full subcategory $\mathcal{S}$ of the category of abelian topological groups. The groups that make up the ...
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$\mathbb Z \oplus \mathbb Z \oplus \mathbb Q \equiv \mathbb Z \oplus \mathbb Q \oplus \mathbb Q$?

Is it true that in signature $\{0,+,=\}$ of abelian groups $\mathbb Z \oplus \mathbb Z \oplus \mathbb Q \equiv \mathbb Z \oplus \mathbb Q \oplus \mathbb Q$? ($ M \equiv M'$ iff $Th (M) = Th (M')$). (...
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1answer
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If $\mathbb Q \otimes_\mathbb Z \mathbb Q \cong \mathbb Q^\mathbb N$, why is $\mathbb Q \otimes_\mathbb Z \mathbb Q$ a $1$-dim $\mathbb Q$-v.s.

In Dummit & Foote, it is an exercise to show that $\mathbb Q \otimes_\mathbb Z \mathbb Q$ is a $1$-dimensional $\mathbb Q$-vector space. This is fairly easy: a $\mathbb Q$-basis for $\mathbb Q \...
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If $H_1$ and $H_2$ are isomorphic normal subgroups of $G$, when do we have an isomorphism between $G/H_1$ and $G/H_2$?

This question is related to the following three questions: Two subgroups $H_1, H_2$ of a group $G$ are conjugate iff $G/H_1$ and $G/H_2$ are isomorphic If $H_1, H_2\leq G$ are such that $H_1\cong H_2$...
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The torsion subgroup of a quotient group of a finitely generated abelian group

Let $G$ be a finitely generated abelian group such that $G/G_t$ has rank $n$ and let $H$ be a subgroup of $G$ such that $H/H_t$ has rank $m$. I want to show that $(G/H)/(G/H)_t$ has rank $n-m$. I ...
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Is there a simple formula for the number of subgroups of index 2 of $\mathbb{Z}_2^n$?

Let $\lambda(n)$ denote the total number of subgroups of $\mathbb{Z}_2^n$ of index $2$ (equivalently, of order $2^{n-1}$), for $n=1,2,3,\ldots$ Is there a neat general formula for $\lambda(n)$ ? For ...
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If $a,b$ are permutations of $S_8$ and $G=\langle a,b\rangle$ and $N=\langle a^2,b^2\rangle$. How can I prove that $G/N$ is abelian?

If I've got a group $G=\langle a,b\rangle$ where $a$ and $b$ are permutation from $S_8$, and a subgroup of $G$ $N=\langle a^2, b^2\rangle$ (which is in the center of $G$). I have to prove that the ...
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2answers
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Confused by quotient group (whats the operation): Show quotient group $GL_n(K)/SL_n(K)$ is abelian.

In my introductory abstract algebra course, the quotient group $G/H$ was defined as $$G/H=\{gH:g\in G\}$$ which is a set of sets. In an exercise, I should show that for the group of invertible ...
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1answer
45 views

Which of the following are true about the group $\Bbb{Z}_6×\Bbb{Z}_9×\Bbb{Z}_{15}/\langle(5,5,3)\rangle$?

Let, $\displaystyle{G=\frac{\Bbb{Z}_6\times\Bbb{Z}_9\times\Bbb{Z}_{15}}{\langle(5,5,3)\rangle}}$ Which of the following are ture about $G$? The given options are- (a) $G$ is cyclic. (b) $G$ is Abelian....
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25 views

Extending a map $\langle a \rangle \to \mathbb{Q}/\mathbb{Z}$ to $A \to \mathbb{Q}/\mathbb{Z}$ for $A$ an abelian group [duplicate]

In an exercise I am doing I have an abelian group $A$ and $a \in A$. I have to find a homomorphism $$ f: \langle a \rangle \to \mathbb{Q}/\mathbb{Z} $$ such that $f(a) \neq 0$ and then extend this ...
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1answer
25 views

How to prove $A \cong \langle a \rangle \oplus A / \langle a \rangle $ for abelian group? [closed]

Let $A$ be an abelian group and $a \in A$. I am trying to prove $$ A \cong \langle a \rangle \oplus A / \langle a \rangle $$ where $\langle a \rangle $ is the subgroup generated by $a$. I would ...
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1answer
41 views

Explicitly describing the subgroups of $\mathbb{Z}^{3}$

I am interested in understanding all the subgroups of $\mathbb{Z}^{3}:=\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$. $\mathbb{Z}^{3}$ a free abelian group of rank three, so all subgroups are free ...
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1answer
52 views

Prove if there is a group whose order is $p^2$,and it is non-abelian, then it is cyclic [closed]

Suppose the non-abelian group G whose order is $p^2$ where p is a prime number, prove it is a cyclic group. My work: there is a $\tau\not=e$ in the group and the order of $\tau$ is either $p$ or $p^...

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