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

learn more… | top users | synonyms

0
votes
2answers
38 views

Q as an additive abelian group has no minimal generating set

$\mathbb Q$ as an additive abelian group has no minimal generating set. I have done this question according to the solution given here. First I took a minimal generating set $S$ of $\mathbb Q$ ...
0
votes
1answer
44 views

Prove that a group with exponent 3 is abelian.

Let be $G$ a group. Is the following statement true? If every $x\in G, x\neq e=1$ has order at most 3 (i.e. $x^3=1$), then $G$ is abelian. I wanted to prove that $xy=yx\ \forall x,y\in G$. $$xy=x1y ...
0
votes
1answer
22 views

proving the identity for subgroups.

What is the best way to prove that if a group is a subgroup of some other group? Or more precisely how to prove that they have common identity element?
0
votes
1answer
25 views

What does it mean for a subgroup $H$ of an abelian group $G$ to be less than or equal to $G$?

I am reading through some linear algebra lecture notes and have come across the following notation: $$K \leq G,$$ where $G$ is an abelian group and $K$ is a subgroup of $G$. What does this notation ...
0
votes
1answer
41 views

Intersection of centralizers is normal?

Let $G$ be an arbitrary group, and suppose that $H=C_G(g_1,\ldots,g_n)$ is also the intersection of all centralizers of finite index in $G$, and furthermore $[G:H]<\infty$. Is it true that $H$ is a ...
1
vote
1answer
10 views

Lifting a decomposition of abelian $p$-groups.

Let $A$ be a finite abelian $p$-group and $x\in A$ an element of order $p$. Assume that have the following exact sequence : $$1\rightarrow \langle x\rangle \rightarrow A\rightarrow B_1\times ...
0
votes
0answers
25 views

Can not see the use of Correspondence Theorem

In Herstein`s proof of Fundamental Theorem of Finite Abelian Groups, I don´t see the use of Correspondence Theorem. It says that exist some subgroup $Q$ of G such that $T=Q/B$. But I`m a bit confused ...
2
votes
1answer
86 views

Direct proof that infinite product of copies of $\mathbb{Z}$ is not projective

It is well-known that the abelian group $$A = \prod_{n=1}^\infty \mathbb{Z}$$ is not free (see, for example this MO question), and that over a PID being free is equivalent to being projective (see ...
0
votes
3answers
31 views

How to proceed in the proof of this statement.

I'm reading the proof of "Fundamental Theorem of Finite Abelian Groups" in Herstein Abstract Algebra, and I've found this statement in the proof that I don't see very clear. Let $A$ be a normal ...
0
votes
1answer
14 views

Commutativity of multiplication of cosets of the commutator subgroup

Take a group $H$ with a non-trivial commutator subgroup, and form the quotient group $H^{ab} = H/H'$. Now, take the cosets of the products of elements $a,b$ and $c,d$: $abH'$ and $cdH'$ in $H^{ab}$. ...
2
votes
1answer
109 views

$G$ contains a normal $p$-subgroup

Let $G$ be a non-abelian finite group with center $Z>1$. I want to show that if $G/Z$ is solvable then $G$ contains a normal $p$-subgroup for some prime $p$ with $p\mid |G:Z|$. $$$$ Since ...
2
votes
0answers
69 views

infinitely $p$-divisible elements in $A\otimes \mathbb{Z}_p$

Let $A$ be a (possibly non-finitely generated) torsion-free abelian group. Suppose that $A$ contains no infinitely $p$-divisible elements, then does the same hold for $A\otimes \mathbb Z_p$, where ...
0
votes
2answers
62 views

Prove a group G is abelian if it satisfies x^2 = x for every x in G

I originally solved this problem by simply noting that x^2 = x implies x=e, so the only element in the group is the identity...but this is wrong. I am now stuck on this idea though and I have tried ...
1
vote
4answers
88 views

Show that the group is abelian

Let $M$ be a field and $G$ the multiplicative group of matrices of the form $\begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix}$ with $x,y,z\in M$. I have ...
1
vote
2answers
38 views

Let $L$ be a subgroup of $\mathbb{Z}^3$ of index $16$. What are the possibilities for $\mathbb{Z}^3 /L$?

Let $L$ be a subgroup of $\mathbb{Z}^3$ of index $16$. What are the possibilities for $\mathbb{Z}^3 /L$? Since $L$ has 16 elements, I think it might be $\mathbb{Z}_{16},\mathbb{Z}_2 \oplus ...
0
votes
1answer
36 views

$G \cong \mathbb{Z}_{p^{n_1}}\times \dots \mathbb{Z}_{p^{n_k}}$. If $p=2$ and $n_1>n_2$, prove that $L(G)\cong \mathbb{Z}_2$.

Let $G \cong \mathbb{Z}_{p^{n_1}}\times \dots \mathbb{Z}_{p^{n_k}}$ be a finite abelian $p$-group, in which $n_1\geq \dots \geq n_k$. Define $$L(G)=\{g\in G \;|\;\alpha(g)=g\; ,\forall \alpha\in ...
1
vote
1answer
15 views

Proof that finite abelian groups are the direct sum of p-parts

A proof of this result was given, however I am having trouble understanding why the part in bold is true. Let $A$ be a non-trivial finite abelian group with $|A|$ having distinct prime divisors ...
0
votes
1answer
15 views

Are there any infinite (virtually) polycyclic groups with lattice orders that are not linear orders?

I am interested in noetherian group algebras, so I am learning about polycyclic groups. Specifically, I want to generalize some ideas that work well with $k[\mathbb{Z}^n]$ utilizing the lattice ...
1
vote
1answer
63 views

Generating sets and independent subsets in abelian groups.

Definition. A set $\{x_1,...,x_n\}$ of non zero elements in an Abelian group is independent if, whenever there are integers $m_1,...,m_r$ with $m_1 x_1 + \cdots + m_r x_r = 0$, then each $m_i x_i$ is ...
1
vote
1answer
60 views

$G$ a non-abelian group of order $p^3$. Show that $Inn(G)$ is abelian.

Let $G$ be a non-abelian group of order $p^3$ for prime $p$. Show that $Inn(G)$ is abelian. The center of $G$, $Z(G)$, is of order $p$ (can be seen in this question). I also know that $G / ...
0
votes
1answer
29 views

Subgroups of a direct sum.

Let G be a finite abelian group and let $G = G_1 + G_2$ where the $G_i$ are cyclic. Add it is a $p$-group if you like. How do I prove that it isn't the case that, for some $H_i$, $G = H_1 \bigoplus ...
3
votes
1answer
32 views

rank of an abelian group and its embedment into vectorspace

I am confused about the rank of an abelian group. In this wiki page, https://en.wikipedia.org/wiki/Finitely_generated_abelian_group , we have that $\mathbb{Z}^n \oplus \mathbb{Z}_{q_1} \oplus \cdots ...
2
votes
1answer
34 views

Is there a strategy for expressing finitely generated abelian group as the direct sum of cyclic groups?

I know that every finitely generated abelian group can be expressed as a direct sum of cyclic groups. I wondering how easily we can find the cyclic groups given an abelian group. Specifically, one of ...
0
votes
0answers
73 views

Problem from Marcus' Number Fields

I have been stuck on the 27th problem of the 2nd chapter from Marcus' Number Fields. In it we're given $G$, a free abelian group of rank $n$ and its subgroup $H$, which is again of rank $n$. Now, ...
1
vote
1answer
73 views

Homomorphisms $\frac{\mathbb{Q}}{\mathbb{Z}} \longrightarrow \mathbb{Q}$

Can someone please show me a concrete example of a group homomorphism $$ \frac{\mathbb{Q}}{\mathbb{Z}} \longrightarrow \mathbb{Q}, $$ if it exists? I apparently cannot find any of it (except for the ...
1
vote
1answer
39 views

Defining Presheaves on Categories

I'm learning about sheaves and sheaf cohomology with the eventually goal of using these tools to study Riemann surfaces. My references are Forster's Lectures on Riemann Surfaces and Iverson's ...
0
votes
2answers
37 views

Pontryagin Dual of a Finite Abelian Group [closed]

Let $M$ be a finite abelian group. I want to show that the Pontryagin dual is a finite abelian group, and in particular I am interested in computing the elementary divisors/invariant factors of it. ...
3
votes
0answers
70 views

Show that $G$ is Abelian if and only if $f: G\times G \to G$ is a homomorphism.

Let $G$ be a group. Let $H$=$G\times G$ be the direct product of $G$ with itself. Define $f: H\to G$ to be $f((g,h))=gh$ for any $(g,h)\in H$. Show that $G$ is Abelian if and only if $f$ is a ...
0
votes
1answer
22 views

Let A be a finite group and P be a normal p sylow subgroup. What is the connection between P and $Tor_p(A)$

Let A be a finite group and P be a normal p sylow subgroup. can there be an element $g \in A$ where $order(g) = p^x$ where x>0 and $g \notin P$ ? what I really try to understand is the connection ...
2
votes
0answers
23 views

What triple “tensor product” is this? Is it just isomorphic to a double tensor product?

Consider the abelian groups $A = \Bbb{Q}^{\times}, B = \Bbb{Q}^{\times}, C = \Bbb{Z}^+$. What if we formed a product like: $A \star B \star C = \text{Free}_{\Bbb{Z}}(A \times B \times C)$ ...
1
vote
1answer
37 views

Finite abelian p-group with an element of maximal order

I want to know, following theorem comes from which book? Theorem . Suppose $G$ is a finite abelian $p$-group and $a \in G$ has maximum order, then there exists a subgroup $K⊆G$ such that: $ \langle ...
1
vote
2answers
76 views

Any group of order $15$ is abelian(without sylow theorem)

Prove that, any group of order $15$ is abelian (without help of Sylow's theorem or its application). What I have done so far is, by class equation we know that $|G|=|Z|+\sum\frac{|G|}{C(a_i)}$. Now ...
2
votes
1answer
32 views

Left adjoint to forgetful from modules to abelian groups

What is the left adjoint to the forgetful functor $U : R-\mathsf{Mod} \to \mathsf{Ab}$? Note here that $R$ is a general ring, not necessarily commutative. I've seen that they define it as $F A = R ...
0
votes
0answers
36 views

Prove that if $A,B,C$ are finite commutative groups and $A\times B\cong A\times C$ then $B\cong C$. [duplicate]

Prove that if $A,B,C$ are finite commutative groups and $A\times B\cong A\times C$ then $B\cong C$. Since $A,B,C$ are finite commutative groups hence we can write $A=\Bbb ...
0
votes
2answers
26 views

Maximal subgroups of Z

In the ring of integers, the only maximal ideals are those generated by the prime elements. Is the same true for the group of integers? Are the only maximal subgroups of integers the ones generated by ...
0
votes
0answers
36 views

On conjugacy classes of involution

Let $G$ be a finite solvable group and $H$ be a normal subgroup of $G$. If all the involution of $H$ lie in a conjugacy class of $G$, then what can we say about the structure of $H$?
-1
votes
2answers
50 views

If K is a finite field, proof that $Gl_n(K)$ is not commutative [closed]

The following property was stated during a lecture in Algebra: If K is a finite field and $n \ge 2$ then $Gl_n(K)$ is a non-abelian finite group. I know how to proof that $Gl_n(K)$ is finite ...
0
votes
2answers
48 views

Show that $o(g_1\cdot g_2)=o(g_1)\cdot o(g_2)$ [duplicate]

Let $G$ be an abelian group, $g_1,g_2\in G$ of finite order ($o(g_1)=m,o(g_2)=n)$ with $(o(g_1),o(g_2))=1$ (relatively prime). Show that $o(g_1\cdot g_2)=o(g_1)\cdot o(g_2)$. I have tried the ...
0
votes
3answers
146 views

Group Theory vs Graph Theory [closed]

I would like to know that, For a given graph can we find an associated finite group? If there are more than one such group, what are the differences and similarities between them? Here ...
2
votes
0answers
43 views

Isomorphic product of finite abelian groups

Suppose $X,Y,Z$ are finite abelian groups with $X \times Y \cong X \times Z$. How to show that $Y\cong Z$? If we assume that we can decompose $Y,Z$ into cyclic groups that are powers of primes, I ...
0
votes
1answer
59 views

Center of a group $G$, when the commutator subgroup has index 2 [closed]

Suppose that $G$ is a finite group, $M=G^{'}Z(G)$, $|\frac{G}{M}|=2$ and there is an element $x\in G$ such that $|C_G(x)|=4$. Is it true that $|Z(G)|>1$ ? ($G^{'}$ is the commutator subgroup and ...
0
votes
3answers
37 views

A finite abelian group must contain an element which is the l.c.m. of the orders of its elements.

Let $G={g_1,...,g_n}$ be a finite abelian group of order $n$ and let $m =$ l.c.m.$(|g_1|,...,|g_n|)$. Since $G$ is finite (without loss of generality) suppose $g_1\cdots g_n = g_1$. We know ...
1
vote
1answer
104 views

Abelian group in short exact sequence

If we have a short exact sequence of continuous group homomorphisms between abelian groups $$0 \rightarrow \mathbb{Z} \oplus \mathbb{Z} \rightarrow X \rightarrow \mathbb{Z} \rightarrow 0,$$ can we ...
1
vote
1answer
36 views

Unique subgroup of index 2 in a finite abelian group.

Suppose $G$ is a finite abelian group, all elements of which are their own inverse. If the order of $G$ is greater than $2$, then prove or disprove that the subgroups of index $2$ in $G$ are not ...
0
votes
0answers
25 views

Property of Abelian Group

I am reading the Handbook of the Mathematical Logic and in the page 8 say: An abelian group $G$ has every element of order $\leq$ n if $G$ is a model of $\forall x [ x = 0 \vee 2x = 0 \vee ...
2
votes
0answers
38 views

Index of a maximal subgroup among normal abelian subgroups

Let $P$ be a $p$-group and $A$ maximal among abelian normal subgroups of $P$. Show that: 1) $A=C_P(A)$. 2) $|P:A|\mid (|A|-1)!$. 1) If $A$ is an abelian normal subgrup of a certain ...
1
vote
1answer
101 views

Doubt about a kernel

I was reading the proof of Lemma 1.25 in this thesis and I thought I understood it, but I think I don't. The thing that I don't see clearly is in page 26 where he is showing that $\textrm{ker}\ ...
3
votes
1answer
40 views

What is $\mathbb Z^2/\text{Im}(\phi)$ isomorphic to in the following case?

Let $\phi:\mathbb Z^2\to\mathbb Z^2$ be the map $(x,y)\mapsto (x+y,2y)$. I need to find $\mathbb Z^2/\text{Im}(\phi)$. My guess is that this is isomorphic to $\mathbb Z_2$ but I am having ...
1
vote
2answers
47 views

The center of a group is an abelian subgroup

Let $(G,\circ)$ be a group and let $Z(G):=\{x \in G : ax=xa \ \forall \ a \in G\}$ be the center of $G$. How can I show that $Z(G)$ is an abelian subgroup of $G$? What I did so far: $Z(G)$ is a ...
-1
votes
3answers
46 views

Submodules of a module with a given property

I am curious about the submodules of a module with a given property. Let $M$ be an $R$-module. If $M$ is a finitely generated are the submodules of $M$ finitely generated? If $R=\mathbb Z$, $M$ ...