Questions tagged [abelian-groups]
For questions about abelian groups, including the basic theory of abelian groups as a topic in elementary group theory as well as more advanced topics (classification, structure theory, theory of $\mathbb{Z}$-modules as related to modules over other rings, homological algebra of abelian groups, etc.). Consider also using the tag (group-theory) or (modules) depending on the perspective of your question.
3,872
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Why is a Sylow 5-subgroup abelian?
For weeks I tried to solve the following question on Brilliant:
Fill in the blank: "Every group of order ___ is abelian."
And these are the possible answers I get: 15, 16, 20, 21, 27.
Using ...
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-2
votes
1
answer
60
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Let $h\in G$. Let $\phi_h : G \to G$ st $g\mapsto hgh^{−1}$. Then $G$ is abelian iff $\phi_h = \operatorname{id}_G$ for all $h\in G$.
Let $G$ be a group and $h\in G$. Consider the map $\phi_h : G \to G$ st $g\mapsto hgh^{−1}$, $G$ is an abelian group if and only if $\phi_h = \operatorname{id}_G$ for all $h\in G$.
I know what it ...
- 19
1
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0
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72
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How to show that a subgroup of a product of finite cyclic group has itself this property?
Let $G$ be a subgroup of a finite product of finite cyclic groups. Is it easy to prove that $G$ is itself a finite product of cyclic groups without appealing to the structure theorem of finite abelian ...
- 1,466
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votes
0
answers
25
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On Ulm invariants of a famous abelian p-group due to Prufer
We know that there is a famous example due to Pruefer, of a countable reduced p-group $G$ of length $\omega + 1$ as following://
Let G be the Abelian group generated by the countable set of ...
0
votes
1
answer
54
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How to find invariant factor of this finitely generated group
I'm trying to solve this problem.
Let $G$ be the set $\Bbb Z^2$ with binary operation defined by
$(x_1,y_1)*(x_2,y_2) = (x_1+x_2, y_1+y_2+x_1x_2)$
Show that $(G,*)$ is a finitely generated abelian ...
- 47
2
votes
1
answer
46
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Finding all the possible orders of elements in a quotient group
Let $A$ be an arbitrary Abelian group and let $H:=\{a\in A\mid\exists b\in A\mid a=b^3\}.$ Prove $H$ is a normal subgroup of $A$. Determine all the possible orders of elements in the group $A/H.$
My ...
- 4,538
2
votes
1
answer
58
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The maximal free abelian subgroup that can be embedded in $GL(n,\mathbb{Z})$
I am stuck on this problem and cannot seem to find a good reason for drawing the required conclusion. The problem is as follows:
Given $SL(n, \mathbb{Z})$ a subroup in $GL(n, \mathbb{Z})$. How can ...
1
vote
0
answers
52
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A group closed under scalar multiplication is a vector space
Suppose $G$ is an abelian group closed under scalar multiplication with elements in the field $F$. Is $G$ always a vector space over $F$?
I have been trying to find a counter-example, but failing. In ...
- 59
1
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2
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69
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Abelian groups with all elements having order $2$
This result is not an exercise, but more so a hunch that I am trying to prove for myself. Let's say that $G$ is a finite group with the property that it is both abelian and every non-identity element ...
- 2,206
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0
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35
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Doubt regarding the generality of an equivalent definition for Abelian Group [duplicate]
I was given a problem as follows:
Let $G$ be a finite group of odd order. If $(ab)^{3}=a^{3}b^{3}$ and $(ab)^{5}=a^{5}b^{5}$ for all $a,b\in G$ then $G$ is abelian.
I was able to show that this is ...
- 73
0
votes
2
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114
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Proving that there is a functor $F: Grp \rightarrow Ab$ s.t $F(G)=G_{ab}$- How to deal with the quotient?
Let $G$ be a group. $f$ a group hom. $f:G\rightarrow H $
I define:
$F(G)=G_{ab}$
and
$F$ : $G$ $\rightarrow$ $G_{ab}$ a homomorphism, where $G_{ab} : = G/[G,G]$ is the abelianization of group $G$....
- 2,696
6
votes
3
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208
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When two infinite direct products $\prod_I\Bbb{Z}$ and $ \prod_J\Bbb{Z}$ are isomorphic?
It is known that two free $\Bbb{Z}$-modules $\bigoplus_{I}\Bbb{Z}$ and $\bigoplus_{J}\Bbb{Z}$ are isomorphic if and only if $|I|=|J|$. Moreover, it is true for any two free module over a commutative ...
0
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1
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51
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Free product and direct product of $\mathbb{Z}$
I'm studying Seifert Van Kampen theorem and I evaluate the fundamental group of torus seen as a quotient space in particular
$$T = [0,1]\times[0,1]/\sim\\
(x,0)\sim(x,1) \\
(0,y)\sim(1,y)$$
With $\pi:[...
- 728
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0
answers
28
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Category of LCAG's "measures" the difference between the integers and the reals: what does this mean?
The Wikipedia article on Locally compact abelian Groups (https://en.m.wikipedia.org/wiki/Locally_compact_abelian_group) has the following excerpt in the Categorical properties section:
Clausen (2017) ...
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3
votes
1
answer
51
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Is multiplication by a scalar an open map in topological groups?
If n is any non-zero integer and G is a topological abelian group, under what conditions is the continuous homomorphism $g \mapsto ng$ open or weakly open?
If this map happens to be open for all n and ...
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0
votes
2
answers
73
views
Row reduction and generating sets of a finite abelian group
Let us suppose that we have a finite abelian subgroup which is generated by 2 elements $A,B$. On the other hand, define the following 3 "actions":
Changing $\{A,B\}$ for $\{B,A\}$.
Changing ...
- 1,912
2
votes
0
answers
76
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The structure of a finitely generated abelian group and its endomorphism ring
This is a question from our past exams. I worked it out. Would anyone kind enough to check the correctness? Thanks! (It occurred several times that I somehow skipped crucial steps or overlooked ...
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2
votes
0
answers
36
views
the non-abelian subgroups of the Lamplighter group
The lamplighter group can be defined by the semidirect product:
$$
L_2=(\mathbb{Z} _2) \wr \mathbb{Z} \cong \bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_{2} \rtimes_\phi\mathbb{Z},$$
where $\phi(1)$ &...
- 1,163
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votes
1
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79
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Why $\operatorname{Hom}(\mathbb Z, \mathbb Z) = \mathbb Z$? [duplicate]
Why $\operatorname{Hom}(\mathbb Z, \mathbb Z) = \mathbb Z$?
Is there a proof for this fact? Also, if I want to understand more the hom, ext, tor and tensor functors, should I study homological algebra ...
- 1,835
1
vote
1
answer
53
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Non-discrete LCAG's whose closed non-trivial subgroups are open
Let G be a locally compact abelian group and suppose that for any non-trivial closed subgroup H of G, H is open. It also makes sense to suppose that G is non-discrete (since in that case this ...
- 692
2
votes
0
answers
189
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The cyclic subgroups of $GL_{3}(\mathbf{F}_{p})$
Let $p$ be an odd prime number and let $\mathbf{F}_{p}$ be the finite field of order $p$.
The group $H$ is a cyclic subgroup of $GL_{3}(\mathbf{F}_{p})$ such that there does not exist an abelian ...
0
votes
0
answers
37
views
kernel and cokernel of multiplication with 2
Let $A$ be an abelian group and let $f_n \colon A \to A, x \mapsto 2^n x$ be a morphism. Suppose the kernel and cokernel of $f_1$ are finite $2$-groups. I want to show that then the kernel and ...
- 417
3
votes
1
answer
97
views
When is the intersection of all open normal subgroups equal to the connected component at the identity?
I've managed to show that for locally compact abelian groups, the connected component at the identity $G_0$ is equal to the intersection of all open subgroups of G, since we have that $$A(G_0) = \...
- 692
1
vote
0
answers
65
views
Quotient of the group $\mathbb{Z}^2 \oplus \mathbb{Z} ^2 \oplus \mathbb{Z} ^2$
Does there exists $x_1$, $x_2$ , $y_1$ , $y_2 \in \mathbb{Z}^2$, such that
$$\frac{\mathbb{Z}^2 \oplus \mathbb{Z} ^2 \oplus \mathbb{Z} ^2 }{ \langle (x_1,y_1,0), (0,x_1,y_1) , (x_2,y_2,0), (0,x_2,y_2) ...
- 1,163
1
vote
1
answer
32
views
Subgroups of self dual groups
Let G be an infinite locally compact abelian group that is isomorphic to its own dual. If H is a closed subgroup of G, is it necessarily true that $H \cong \widehat{G/H}$?
I ask because in the case of ...
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7
votes
1
answer
125
views
Finding a set of representatives that sum to zero.
Let $G$ be an abelian group of order $2n$, where $n$ is an odd integer (For simplicity, we may assume $G\simeq \mathbb{Z}_{2n}$ (the cyclic group of order $2n$). Let $A_1,A_2,\ldots,A_n$ be a ...
- 2,815
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votes
1
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138
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Is there a type of product of groups where $C_2 \star C_2 = C_4$?
Hi I am wondering if there is a type of product $\star$ of groups to get $C_4 \cong C_2 \star C_2$.
We know that the composition series of $C_4$ indeed is a 2 copy of $C_2$ so I am wondering if there ...
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0
answers
22
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2-groups with abelianization of type (2,2)
On Olga Taussky's article "A Remark on the class field tower" there is a note on the last page where she states that P. Hall pointed out to her that exists 3 different groups of order $2^k$ ...
3
votes
1
answer
58
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Decomposition of infinite abelian groups
A colleague recently mentioned a "Prüfer Decomposition Theorem", claiming that every abelian group $A$ could be expressed as the direct sum $A = T(A) \oplus F(A)$, where $T(A)$ is the ...
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1
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45
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Why Scalar Multiplication is not commutative/Abelian?
Definition 2.9 (Vector Space). A real-valued vector space V = (V,+, ·) is vector space
a set V with two operations
: V × V → V (2.62)
· : R × V → V (2.63)
where
(V, +) is an Abelian group
Source: ...
- 21
1
vote
0
answers
15
views
Groups extensions by linear cocycles
Let $0 \to A \to X \to C \to 0$ be an abelian group extension (we require $X$ to be abelian too). Then the group operation on $X$ is described by a $2$-cocycle $c(x, y) = s(x) + s(y) -s(x+y)$ where $s:...
- 546
2
votes
0
answers
117
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Is it possible to write a code that computes all the subgroups (up to isomorphism) of a finite Abelian group?
I am new to computational group theory. I am trying to find/write a program that computes the following:
Input: a finite abelian group $$G \cong \mathbb{Z}_{m_1} \oplus \cdots \oplus \mathbb{Z}_{m_{...
- 1,163
6
votes
0
answers
79
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Abelian subgroups of the group of automorphisms of a finite group
This is a follow-up question from my post here, which has been moved according to a comment. For context, here is the setup.
Let $G$ be a nontrivial finite group. In his book "Finite Group ...
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3
votes
1
answer
66
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Size of $p$-subgroups of $\operatorname{Aut}(G)$, where $p$ divides the order of $G$
Let $G$ be a nontrivial finite group. In his book "Finite Group Theory", M. Isaacs proves the following two results:
Corollary 3.3: Let $\sigma \in \operatorname{Aut}(G)$. Then, $o(\sigma) &...
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1
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Given two isomorphic subgroups of a finite abelian group with the same number of minimal generators, are the quotients isomorphic?
Let $G$ be a finite abelian group.
We know that the least size of a minimal generating set of a proper nontrivial subgroup $H \leqslant G$ might be the same as $G$, for example ($\Bbb Z_6 \cong \...
- 1,163
0
votes
0
answers
24
views
Spectral sequences of bicomplexes: are differentials chain complex maps?
This is a probably stupid question, and I guess the answer is no, but it's worth trying.
Consider the spectral sequence $E^{pq}_r$ associated to a bicomplex (of abelian groups). At page $r$ we will ...
- 5,224
0
votes
0
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11
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Maximal Essential Extension in Finitely Generated Abelian groups
In GTM4, the author asked the reader to give a procedure for calculating the maximal essential extension of $A$ in $B$, where $B$ is a finitely generated abelian group.
I've got completely no idea of ...
- 101
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0
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14
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Maximal Essential Extension of Abelian Groups
Given any abelian group $A$, let $T(A)$ be its torsion subgroup and $F(A)=A/T(A)$. A homomorphism $\varphi:A\to B$ induces $T(\varphi):T(A)\to T(B)$, $F(\varphi):F(A)\to F(B)$, and $\varphi$ is one-to-...
- 101
9
votes
1
answer
252
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Weird duality between integers and p-adic integers
The integers are, up to isomorphism, the unique infinite discrete abelian group that is isomorphic to all its non-trivial subgroups. This can be seen easily by Pontryagin Duality: It's equivalent to ...
- 692
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0
answers
29
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Preimage of a canonical vector with a coprime coordinate in a epimorphism of abelian groups
Let $s,r$ be natural numbers such that $s \geq r$, $$B = \prod_{i = 1}^r\mathbb{Z}_{e_i}$$
a finite abelian group where $1 < e_r$ and $e_{i+1} | e_i$ for all $i$ and consider $\phi$ a surjective ...
- 313
1
vote
2
answers
72
views
Given $\text{Hom}(M,M')$ is abelian, does $M$ have to be abelian?
I am doing a course on algebra (self-study) and it was left as an excercise to prove that if $M$ and $M'$ are abelian groups, then $(\text{Hom}(M,M'), +)$ is an abelian group, where the operation is ...
- 1,451
2
votes
0
answers
44
views
What is $H^2(U(1),\mathbb Z_2)$?
Let $U(1)$ denote the multiplicative group of complex numbers of modulus $1$ and $\mathbb Z_2:=\mathbb Z/(2\mathbb Z)=\left(\{0,1\},+\right)$
I know that the central group extension
$$\mathbb Z_2\...
- 1,828
0
votes
1
answer
90
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Abelian, transitive subgroup of $S_A$, then $\sigma(a)\neq a$ for $\forall\sigma \in G-\{1\}$ and $\forall a$
I have read these questions:
Show that $\sigma(a)\ne a,\forall\sigma\in G-\{1\}$ and all $a\in A$.where $G$ is abelian, transitive subgroup of $S_A$,
Show that any abelian transitive subgroup of $S_n$ ...
- 1,069
-1
votes
1
answer
70
views
Basis of tensor product over non-commutative ring. [closed]
Let $M$ and $N$ be $R$-modules, where $R$ is a ring, and let $M\otimes_R N$ be the tensor product over $R$. If $R$ is commutative, then the basis of $M\otimes_R N$ is the set
$$
\{ m\otimes_R n :m\in ...
- 833
1
vote
0
answers
54
views
A "simpler" description of the automorphism group of the Lamplighter group
The lamplighter group is defined by the following presentation:
$$
L_N=(\mathbb{Z} _N) \wr \mathbb{Z} \cong\left\langle t, a_n, n \in \mathbb{Z} \mid a_n^N, t a_n t^{-1}=a_{n+1}, n \in \mathbb{Z}, a_n ...
- 1,163
1
vote
0
answers
80
views
What is ${\rm Aut}\bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_p$?
Let $p$ be a prime, what is the automorphism group of $G=\bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_p$ (countable infinite direct sum of $\mathbb{Z}_p$)?
I know every permutation of the coordinates will ...
- 1,163
0
votes
0
answers
29
views
$\prod_p \mathbb{Z}/p\mathbb{Z}$ is not the direct sum of $\bigoplus_p \mathbb{Z}/p\mathbb{Z}$ and a torsion-free subgroup
While I was reading "Abelian Groups" by Fuchs $(2015)$, I encountered Example $1.2$ in the chapter on Mixed Groups, which stated the following:
Let $p_1,p_2,\dots,p_n,\dots$ denote different ...
- 69
0
votes
0
answers
38
views
Comparing spectral sequences with different coefficients
Several spectral sequences coming from geometry have a "degree of freedom" in choosing the coefficients (say an abelian group $A$). This applies fo example to the Serre spectral sequence and ...
- 5,224
5
votes
1
answer
214
views
Real Elliptic curves as compact abelian groups
It's well known that an elliptic curve of the form $y^3 = x^3 +ax +b$ admits a group structure, as long as a point at infinity $O$ is added to serve as identity. If we look at a real elliptic curve as ...
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3
votes
0
answers
54
views
$\underset{p\in \mathbb{P}}{\prod}\mathbb{Z}/p\mathbb{Z}$ is non splitting mixed abelian group.
We say that an abelian group $G$ is mixed if it has elements $ \neq 0$, that are of finite order (torsion elements), as well as elements of infinite order (torsion-free elements). We denote the ...
- 69