Tagged Questions
8
votes
1answer
56 views
Cyclic subgroups of $GL_2(\mathbb{Z}/p\mathbb{Z})$?
What are the cyclic subgroups of $GL_2(\mathbb{Z}/p\mathbb{Z})$, the general linear group over the finite field of order $p$, where $p$ is prime?
Obviously, each cyclic subgroup is generated by some ...
0
votes
1answer
61 views
4
votes
2answers
59 views
If the order of a finite abelian group is not divisible by a square, show that the group must be cyclic.
If the order of a finite abelian group is square free, show that the group is cyclic.
This is a question from "basic abstract algebra" by bhattacharya
2
votes
2answers
92 views
Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$.
Suppose $G$ is an abelian group and $a\in G$ and
$$f:\left<a \right>\to\Bbb T$$
is a homomorphism. Can $f$ be extended to a homomorphism on $G$:
$$g:G\to \Bbb T$$
?
$\Bbb T$ is the circle ...
2
votes
0answers
21 views
Separable elements of a finite abelian group
Let $\mathbf G$ be a finite abelian group, let $a, b \in \mathbf G$, and let $\langle a \rangle$ and $\langle b \rangle$ be the cyclic subgroups of $\mathbf G$ generated by $a$ and $b$ respectively.
...
3
votes
1answer
98 views
Finite Abelian groups, G, H, K: $G \times H \cong G\times K$ then $H\cong K$
Let $G,H,$ and $K$ be finite abelian groups then if $G \times H \cong G\times K$ then $H\cong K$.
I am trying to use the fundamental theorem for abelian groups to solve this, it is clear intuitively ...
2
votes
0answers
67 views
$G$ fin ab group, acts faithfully, transitively on $X$, then $|X|=|G|$
Let $G$ be a finite abelian group. Suppose that $G$ acts faithfully and transitively on a set $X$. Show that $|X|=|G|$. Deduce that the action is equivalent to the action of $G$ on itself by left ...
0
votes
3answers
56 views
Groups - Prove that if $G/Z(G)$ is cyclic then $G$ is abelian
Prove that if $G/Z(G)$ is cyclic then $G$ is abelian. Using this fact and $G$ is a nontrivial group of prime power order, deduce that a group of order $p^2$ , $p$ prime, is abelian.
1
vote
3answers
101 views
Why doesn't the Chinese remainder theorem contradict the Fundamental Theorem of Finitely Generated Abelian Groups?
I am finding a contradiction between those two theorems and I do not know what I am doing wrong. First theorem is:
The group $\Bbb Z_{m_1} \times \Bbb Z_{m_2} \times \dotsm \times \Bbb Z_{m_n}$ is ...
7
votes
1answer
92 views
on finite abelian groups
Let $G$ be a finite abelian group and let $M(G)$ be the set of all elements of $G$ that fix with any automorphism of $G$. Then prove
$$M(G)=\langle1\rangle \text{ or } Z_{2}$$
Attempt: We know that ...
2
votes
3answers
91 views
Show that for G be a abelian group, $g \in G, g^m=1$, where $m=|G|$
I am pretty sure there is a name for the following theorem, but unfortunately, I don't known it.
Theorem. Let $G$ be a abelian group with $m=|G|$, than for any $g \in G, g^m=1$.
Currently I don't ...
3
votes
1answer
36 views
Maximal Subgroups Containing given Element
Let $G$ be an elementary abelian $p$-group of finite rank, and $1\neq g\in G$. How do we parametrize the maximal subgroups of $G$, which contain $g$?
5
votes
5answers
121 views
What is the number of distinct homomorphism from $\Bbb Z/5 \Bbb Z$ to $\Bbb Z/7 \Bbb Z$
What is the number of distinct homomorphism from $\Bbb Z/5 \Bbb Z$ to $\Bbb Z/7 \Bbb Z$ and how to find it?
I came across the above problem and do not know how to get it? Can someone point me ...
1
vote
5answers
152 views
Find all the subgroups of $\mathbb{Z}_7$ and $\mathbb{Z}_9^\times$
Can you please help me in this question:
Find all the subgroups of $\mathbb{Z}_7$ and $\mathbb{Z}_9^\times$.
Thanks a lot
1
vote
3answers
59 views
Formula for Product of Subgroups of $\mathbb Z$, Problem
What is the product of $\mathbb{Z}_2$ and $\mathbb{Z}_5$ as subgroups of $\mathbb{Z}_6$?
Since $\mathbb{Z}_n$ is abelian, any subgroup should be normal. From my understanding of the subgroup product, ...
6
votes
2answers
90 views
If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.
Assume that $\pi$ is an isomorphism of a finite group $G$. Let $S$ denote the set $\lbrace g \in G: \pi (g)=g^{-1} \rbrace$. Show that if $|S|>\frac{3|G|}{4}$, then $G$ is an abelian group. Anyone ...
7
votes
1answer
75 views
Finding the order of the automorphism group of the abelian group of order 8.
So I am given an abelian group of order $8$ such that for all non-identity elements $x^2 = e$ (all elements have order two). So I know the answer is gonna be $168$, but I gotta prove this.
So far I ...
1
vote
3answers
113 views
On Groups of Order 315 with a unique sylow 3-subgroup .
in dummit and foote , an exercise asked me to prove that , if$ G$ is a group of order $315$ , $G$ has a normal sylow $3$-subgroup then , $G$ is abelian .
this is exercise number $27$ , section $5$ , ...
6
votes
2answers
164 views
Facts about Abelian Groups and group order.
I look for some theorems which tell us about the relation between the property of being abelian for groups and the order of the group.
I think these theorems are provided in a second course of group ...
3
votes
2answers
67 views
$\pi$-radical of group
Let $G$ abelian and periodic. Let $\mathbb{P}$ be the set of prime numbers, $\pi
\subseteq \Bbb P$ and $\pi ^{\prime }=\Bbb P\setminus\pi $.
Let $O_{\pi }\left( G\right) =\left\langle ...
2
votes
1answer
44 views
Conjugate of an abelian maximal subgroup is maximal
Perhaps this is a trivial question. Let $G$ be a finite group, and $M$ be a maximal (proper) subgroup of $G$. Suppose also that $M$ is abelian. How could I prove that if $x\in G$, then $xMx^{-1}$ is a ...
0
votes
2answers
91 views
Existence of a subgroup
Let $G$ be a finite abelian group and $H$ a subgroup of $G$, then exists a subgroup $L$ of $G$ such that $L≃G/H$
1
vote
2answers
78 views
Generators of fields, extending groups to fields, finite abelian groups
So I'm working through Koblitz'z "a course in number theory and cryptography" when I came across his proof that every finite field has a generator (ie, There is an element such that the multiplicative ...
9
votes
2answers
116 views
The existence of a group automorphism with some properties implies commutativity.
Let $G $ be a finite group, $T$ be an automorphisom of $ G $ st $ Tx = x \iff x=e $. Suppose further that $ T^2 =I $. Prove that $ G $ is abelian.
I was thinking if I show $ T aba^{-1} b^ ...
3
votes
4answers
357 views
Group tables for a group of four elements. [duplicate]
I should consider group tables obtained by renaming elements as essentially the same and then show that there are only two essentially different groups of order 4.
There seems to be so many different ...
-3
votes
1answer
236 views
prove : a finite abelian group G has a subgroup of order d for all d dividing the order of G
Use Cauchy’s Theorem and induction to prove that a finite abelian group G has a subgroup of order d for all d dividing the order of G .
if there is other proof without
2
votes
3answers
59 views
abelian and finite group
G= $Q^+$ (Rational numbers diffrent from zero)
$a*b = ab/2$
I already proved this is a group now I need to prove or disprove that it is abelian and or finite group.
For abelian - from what I ...
0
votes
1answer
51 views
Group of order three distinct primes
I want to show that a group G of order 345 is Abelian.
I used Sylow's theorem to find Syl(5)=Syl(23)=1. but i was unable to conclude Syl(3)=1 because i found Syl(3)=1 or 115. I'm not sure how to ...
2
votes
2answers
84 views
Abelian groups of order $14,27,30,$ and $21$.
Which of the following statements is false?
Any abelian group of order $27$ is cyclic.
Any abelian group of order $14$ is cyclic.
Any abelian group of order $21$ is cyclic.
Any abelian group of ...
1
vote
1answer
143 views
How do you Classify all such groups?
Assume that:
$G$ contains a normal subgroup $H$ of order $9$, and $G$ is generated by $H$ and an element $x\in G-H$ of order $3$.
How to classify all such groups $G$?
I think $9$ divides the ...
1
vote
2answers
151 views
Find number of abelian groups of order $27$?
I was trying to solve the following problem:
Find number of abelian groups of order $27$ ?
Could someone point me in the right direction?
Thanks in advance for your time.
4
votes
1answer
107 views
Normal Abelian Subgroups with Relatively Prime Indices
Suppose A, B are normal abelian subgroups of some finite group G. Let [G:A]=m, [G:B]=n, where gcd(m,n)=1.
Can G be non-abelian?
--
I've been attempting to show that G must be abelian, but I'm ...
1
vote
2answers
136 views
Isomorphisms between finite abelian groups
I am working on the following problem:
Are there any isomorphisms between the following finite abelian groups?:
$$\mathbb{Z}_{1225}\times\mathbb{Z}_{315},$$ ...
0
votes
1answer
51 views
$G/G' \cong I/I^2$ where $I$ is the augmentation ideal [duplicate]
Possible Duplicate:
Isomorphism between $I_G/I_G^2$ and $G/G&#39;$
Let $G$ be a finite group. Let $I\unlhd\mathbb{Z}[G]$ be the augmentation ideal.
I'm trying to prove that
...
2
votes
2answers
177 views
Isomorphic to the Klein 4 group
I need to prove the group of the units of $\mathbb{Z}_3\times\mathbb{Z}_3$ is isomorphic to the Klein-4 group. But I'm really struggling to prove this. Any hints to start me off in the right ...
15
votes
3answers
344 views
How “abelian” can a non-abelian group be?
Something I have been wondering: in general, is there a bound for how many elements in a finite non-abelian group $G$ can commute with every other element? Equivalently, is there is a bound for the ...
2
votes
1answer
93 views
$G$ finite group, $H \trianglelefteq G$, $\vert H \vert = p$ prime, show $G = HC_G(a)$ $a \in H$
Let $G$ be a finite group. $H \trianglelefteq G$ with $\vert H \vert = p$ the smallest prime dividing $\vert G \vert$. Show $G = HC_G(a)$ with $e \neq a \in H$. $C_G(a)$ is the Centralizer of $a$ in ...
7
votes
3answers
113 views
Subgroups of $\mathbb{Z}_2 \times \mathbb{Z}_{12}$ of order $6$
what are the subgroups of $\mathbb{Z}_2 \times \mathbb{Z}_{12}$ of order $6$? I know that there are three such subgroups,
and two subgroups are clear to me, namely the subgroup isomorphic to ...
5
votes
3answers
180 views
$G$ group, $H \trianglelefteq G$, $\vert H \vert$ prime, Prove $H \leq Z(G)$,
Let $G$ be a finite group. Let $H \trianglelefteq G$, with $\vert H \vert = p$, a prime, where $p$ is the smallest prime dividing $\vert G \vert$. Prove: $H \leq Z(G)$. (Hint: If $a \in H$, by ...
3
votes
1answer
105 views
The cancellation property for finite abelian groups
I need some hints to prove that:
Let $A,B,C$ are finite abelian groups such that $A\oplus B\cong A\oplus C$. Prove that $B\cong C$.
I know that every finite abelian group can be written as a ...
1
vote
1answer
73 views
Seeking a proof of: If any two Abelian groups of order $d$ are isomorphic, then $d$ is squarefree.
Suppose that $d \in \mathbb{N}$ satisfies the property that given any two Abelian groups $A_1, A_2$ of order $d$, $A_1 \cong A_2$. Prove that given any prime $p$, $p^2 \nmid d$. What can one say for ...
0
votes
1answer
63 views
showing G is abelian
If $G$ group of order $52$ includes a normal group of order $4$ then $G$ is abelian.
I did like this
$$
|G|=52=2^2\cdot 13
$$
let $H$ be normal group of order $4$.
$n_{13}=1$ thus $G$ has a $K$ ...
19
votes
1answer
432 views
Recovering a finite group's structure from the order of its elements.
Suppose you know the following two things about a group $G$ with $n$ elements:
the order of each of the $n$ elements in $G$;
$G$ is uniquely determined by the orders in (1).
Question: How ...
2
votes
1answer
90 views
p-group: cyclic $n \leq 1$ | abelian $n \leq 2$
$p$ prime number, $n$ a non-negative integer, $G$ group.
(a) $\forall G$ with $|G| = p^{n}$ cyclic $\Leftrightarrow$ $n \leq 1$.
(b) $\forall G$ with $|G| = p^{n}$ abelian $\Leftrightarrow$ $n \leq ...
1
vote
2answers
52 views
inverse of even number of elements in a group
if an abelian group with |G|=n where n is odd. if i take out the identity i'm left with even # of distinct elements. can this mean that each element has an inverse which is not itself??
not a homework ...
0
votes
1answer
143 views
Groups of the same order that are nonisomorphic
I'm reading A First Course in Abstract Algebra by Fraleigh and I've reached a point where I feel like I'm supposed to have understood something more from the chapter than what is actually stated. I've ...
4
votes
1answer
124 views
Smallest pure subgroup containing a fixed subgroup
I will ask a slightly more precise question then in the title.
Let $G$ be a finite abelian group, and $g_1, \ldots, g_n \in G$ such that the cyclic groups they generate are in direct sum $\langle g_1 ...
3
votes
3answers
165 views
Splitting exact sequences of finite abelian groups
I would like to find a condition for an exact sequence of abelian groups
$$
0\to H\to G\to K\to 0
$$
to split. Assume for simplicity that $H=\langle h \rangle$ is cyclic, and choose a basis for $G= ...
3
votes
3answers
98 views
Constructing a basis for finite abelian groups
Let $G$ be a finite abelian group, and $g_1, \ldots, g_k$ a set of "linearly independent elements", namely such that $\langle g_1 \rangle \oplus \ldots \oplus\langle g_k \rangle$.
I would like to ...
3
votes
1answer
138 views
Linear algebra of finite abelian groups
Let $\phi:G \to H$ be a surjective homomorphism of finite abelian groups, and
let $g_1, \ldots, g_n$ be an irredundant set of generators (from now on, a basis) for $G$. be a basis for $G$, meaning a ...
