All Questions
73 questions
-1
votes
1
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203
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Every group of order p^2 is abelian. [closed]
I am studying for my algebra qualifying exam in January, and I have a question regarding the proof I have for this question in Hungerford, problem 13 under the Sylow Theorems section.
The way I ...
4
votes
2
answers
169
views
Let $p$ and $q$ be twin primes, both greater than or equal to $5$ , is every group of order $p^2q^2$ abelian?
I don't know if I made a mistake, but it seems that something is missing here.
Let $p$ and $q$ be twin primes, i.e., $q = p + 2$. We want to prove that every group with order $p^2q^2$ is abelian, ...
- 153
3
votes
2
answers
174
views
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 ...
- 444
0
votes
2
answers
343
views
Let $G$ be a group of order $30$. Show that if $G$ is nonabelian, there are more than one $2$-subgroups of Sylow.
Let $G$ be a group of order $30$. Show that if $G$ is nonabelian, there are more than one $2$-subgroups of Sylow.
Suppose there is only one $2$-subgroup of Sylow and show that $G$ is abelian.
It is ...
- 873
2
votes
1
answer
112
views
Counting the number of elements of order $p$ in an abelian group if $p$ divides the order of the group?
If $G$ is abelian, then the Sylow $p$-subgroups are unique, right (because they're normal)?
That should give the number of elements of order $p$ equal to $n_p(p-1)=p-1$. But that's the wrong answer. ...
- 989
3
votes
2
answers
63
views
Why does $P_{17}$ being abelian imply that $P_{17} \leq N_G(P_3)$?
It is not clear to me in the first answer to this question in the last two lines why $P_{17}$ is abelian implies that $P_{17} \leq N_G(P_3)$ and what is the contradiction that we obtain that leads to $...
- 2,197
2
votes
0
answers
29
views
Is every subgroup of a finite abelian $p$-group $G$ arisen from some decomposition of $G$? [duplicate]
Is every subgroup of a finite abelian $p$-group $G$ arisen from some decomposition of $G$?
To be specific, given a finite abelian $p$-group $G$ and pick any subgroup $H \leq G$. Is there always some ...
- 1,103
1
vote
3
answers
96
views
Cyclic groups and abelian groups [duplicate]
Why abelian group of order 30 is always cyclic? I know that in a group of order 30 either subgroup of order 3 or subgroup of order 5 is normal
3
votes
0
answers
184
views
A proof for the converse of Lagrange's Theorem (for finite abelian groups).
I must follow two instructions to prove the converse of Lagrange's Theorem for finite abelian groups.
First I must prove $(\star)$
Let $q_1,\dots,q_n\in\mathbb{N}$ and $k\in\mathbb{N}$ such that
$k\...
- 639
0
votes
1
answer
124
views
If Cylic subgroup implies abelian implies normal then how A5 is simple group [closed]
I am facing problem $A_5$ is simple group, but $A_5$ had 10 cyclic subgroup of order 3, from cyclic $\Rightarrow$ abelian $\Rightarrow$ normal we can say $A_5$ has 10 normal subgroups, but $A_5$ is ...
0
votes
0
answers
59
views
How to prove an abelian group $G$ must have an element of $p$, $p$ is a prime divisor of $|G|$ [duplicate]
An exercise:
Assume $G$ is a finite abelian group, and let $p$ be a prime divisor of $|G|$. Prove that there exists an element in $G$ of order $p$. (Hint: Let $g$ be any element of $G$, and consider ...
- 41
2
votes
2
answers
155
views
Let $p,q$ be primes and $|G|=p^3q^3$ for an abelian group $G$. Prove that there exists a subgroup $H$ of order $pq^3$.
Let $p,q$ be primes and $|G|=p^3q^3$ for an abelian group $G$. Prove that there exists a subgroup $H$ of order $pq^3$.
The hint tells I can use Cauchy, Sylow and Factor Theorem, so that there is a ...
- 39
1
vote
0
answers
125
views
Every group of order 725 is abelian
I had a test in representation theory today and one of the questions regarded a non-abelian group of order 725. However, I think I managed to prove that every group of order 725 is abelian. If I have ...
- 59
2
votes
0
answers
87
views
Prove that if $G$ is a group of order $315$ with a normal sylow-3 subgroup is abelian. [duplicate]
Prove that if $G$ is a group of order $315$ with a normal sylow-$3$ subgroup is abelian.
My approach:
Let $|G| = 3^2.5.7$.
If we can show that $Z(G)$ contains the sylow $3$ subgroup then $|G/Z(G)|=5,...
- 1,982
0
votes
0
answers
48
views
Let $G$ be finite s.t. there exists an epimorphism $\varphi: G\to\Bbb{Z}_2\times\Bbb{Z}_2\times\Bbb{Z}_3$. Show $G$ has no cyclic Sylow $2$-subgroup.
Let $G$ be a finite group such that there exists an epimorphism $\varphi: G \to \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3$. Show that $G$ has no cyclic Sylow $2$-subgroup.
Attempt: If I ...
- 6,653
4
votes
3
answers
217
views
Proving that group $G$ of order $|G|=35$ is Abelian
This is my outline of proof:
By Sylow's theorems, $G$ has two unique subgroups $H$ and $K$ respectively of order $5$ and order $7$ and both are Abelian; as groups of prime order are Abelian
Next I ...
- 405
0
votes
1
answer
67
views
Showing that $G$ is abelian, using order of $G$ and center of $G$.
My question is the following: Let $G$ be a group of order $7 \times 43 \times 47$ and $Z(G)$ contains an element of order $7$. Show that $G$ is abelian.
Note: This is a question from a algebra qual ...
- 98
1
vote
0
answers
69
views
Minimal normal subgroup contained in a normal Sylow $p$-subgroup is abelian
I am studying the classification of sharply k-transitive groups, and I am reading a proof that sharply $2$-transitive groups are contained in an affine group: Aff($GF(p)$).
One of the steps it makes ...
- 1,104
1
vote
1
answer
350
views
Let $|G|=80$ be an abelian group. Find the number of elements of order $20$ in $G$ given there's more than $33$ of those, and no element of order $40$
Let $G$ is an abelian group of order $80$, given that there is more than $33$ elements of order $20$, and given $G$ doesn't have an element of order $40$, find the number of elements of order $20$.
...
- 2,817
3
votes
2
answers
220
views
The center of a group of order $3^2q$ has order $1$ or $3$.
this is a question from a brazillian book, Paulo A. Martin's "Grupos, Corpos e Teoria de Galois" (portuguese for "Groups, fields and Galois Theory" (The actual word field translate ...
3
votes
2
answers
76
views
If abelian $P\in{\rm Syl}_p(G)$, and $H\le P$ and $H^g\le P$, show $g\in G$ is the left product of an element of $N_G(P)$ with an element of $C_G(H)$
If abelian $P \in{\rm Syl}_p(G)$, and $H \leq P$ and $H^g \leq P$, show $g \in G$ is the left product of an element of $N_G(P)$ with an element of $C_G(H)$
First, I set $g=nc$, where I hope to find ...
- 241
1
vote
1
answer
202
views
Doubts on proving there are only $2$ abelian groups of order $12$, up to isomorphism?
I have been tasked with the following exercise:
Prove there are only $2$ abelian groups of order $12$, up to isomorphism.
I have read some texts and watched some videos but got very confused. I want ...
- 24.5k
1
vote
1
answer
735
views
Let $G$ be a group of order 77. Show that $\operatorname{Aut}(G)$ is abelian but not cyclic.
The proposition is this:
Let $G$ be a group of order 77. Show that $\operatorname{Aut}(G)$ is abelian but not cyclic.
There's also a hint to use Sylow theorems for both $G$ and $\operatorname{Aut}(G)$....
5
votes
0
answers
358
views
Bounds for probability that two elements commute in a group?
Let $G$ be a finite non-abelian group, and lets randomly choose two elements of $G.$ It seems quit well known that the probability that they commute is at most $\text{Pr}(G)\le\dfrac{5}{8}.$ Here is a ...
- 18.7k
6
votes
1
answer
900
views
(Dummit and Foote) Group of order 105 with $n_3 = 1$ must be abelian
I was working on this problem: Let $G$ be a group of order $105 = 3\times 5\times 7$. Assume it has a unique normal Sylow 3-subgroup. Then prove that $G$ is abelian.
I worked out the following from ...
user581023
0
votes
1
answer
245
views
Isomorphic Sylow p-subgroups of two finite abelian groups G and H
Let $G$ and $H$ be abelian groups of order $n$. I want to prove that $G$ is isomorphic to $H$ if and only if for every prime $p\mid n$, Sylow $p$-subgroup of $G$ is isomorphic to Sylow $p$-subgroup of ...
0
votes
1
answer
66
views
Show that the group in the center of this short exact sequence is abelian
Suppose that $$1\to A\xrightarrow{f} B\xrightarrow{g}C\to 1$$ is an exact sequence of groups, where $A$ has order $85$ and $C$ has order $9$. Show that $B$ is abelian.
I have proved earlier that for a ...
- 800
0
votes
1
answer
739
views
A simple proof of Sylow theorem for abelian groups
In my attempt to prove
Let $G$ be an abelian group and $H,K$ its subgroups such that $o(H) = m$ and $o(K) = n$. Then $G$ has a subgroup of order $\operatorname{lcm}(m,n)$.
, the crucial argument is
...
- 17.9k
5
votes
1
answer
2k
views
A finite abelian group is isomorphic to the direct product of its Sylow subgroups
I've seen this question here before, but I want to know if the following is sufficient:
Attempt:
First note that the product of two normal subgroups $H_1$ and $H_2$ is itself a normal subgroup, and ...
- 744
8
votes
2
answers
4k
views
Group of order 45 is abelian
I am trying to prove that the inverse of Lagrange Theorem holds for a group $|G|$ of order $45$ and that it contains a subgroup of order $3$. I think I proved that $G$ is abelian and therefore the ...
- 761
4
votes
3
answers
5k
views
Is a group of prime-power order always abelian?
Let $G$ be a group of order $p^n$, with $p$ prime. By Sylow's first theorem, there exists at least one subgroup of order $p^n$ (the number of subgroups of order $p^i$ is $1$ mod $p$ per $i$). The ...
- 391
1
vote
1
answer
218
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 ...
- 3,142
0
votes
0
answers
50
views
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-...
- 4,747
3
votes
1
answer
292
views
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, ...
- 623
1
vote
2
answers
2k
views
Number of elements of order $2$ in a group of order $10$.
Consider a group $G$ of order $10$.
Then $G$ can be abelian as well non-abelian.
What is the number of non-trivial elements of $G$ of order $2$?
Answer:
If $G$ is abelian, $G$ can be cyclic as ...
- 10.8k
2
votes
1
answer
397
views
Prove that if $H$ and $K$ are Sylow $p$-subgroups, then $H=K$.
Let $G$ be a finite abelian group and let p be a positive prime number that divides the
order of $G$. Prove that if H and K are Sylow p-subgroups, then $H = K$.
I'll first assume that $H$ and $K$ are ...
- 464
0
votes
1
answer
48
views
For which primes $l$ are the $l$-Sylow subgroups of $\Bbb Z_{p^3q} \times \Bbb Z_{p^2} \times \Bbb Z_{q^2r}$ cyclic?
I am trying to find the cyclic Sylow $l$-subgroups of $A = \Bbb Z_{p^3q} \times \Bbb Z_{p^2} \times\Bbb Z_{q^2r}$, where $p,q,r$ are prime numbers.
I have found the elementary divisor composition of ...
- 21
1
vote
0
answers
87
views
Prove that a group of order 45 is necessarily abelian by Sylows' Theorems [duplicate]
So far, my attempt of proof looks like this:
$\#G =45=3^2\times 5$, so:
By the 1st Sylow Theorem that G has p-subgroups of order 5, 9 and 3.
By the 3rd theorem, the p-subgroups of order 5 and 9 have ...
- 442
2
votes
0
answers
245
views
Simple proof of Sylow's theorems for Abelian groups?
This question considers a proof of Sylow's first theorem (for every prime factor $p$ of $|G|$, a Sylow $p$-subgroup of $G$ exists).
My book proves this in the following way: prove the theorem for ...
- 7,606
2
votes
1
answer
322
views
If $q$ not congurent to $1 \pmod{p}$, then every group of order $p^2q$ is abelian.
Let $p$ and $q$ be prime numbers with $p<q$.
If $q \not\equiv 1 \pmod{p}$, then every group of order $p^2q$ is
abelian.
From the Third Sylow Theorem I found that the number of $p$-Sylow subgroups ...
- 177
0
votes
1
answer
661
views
Non-abelian group of order 75 must have subgroup $P \cong Z_5 \times Z_5$
Let $G$ be a non-abelian group of order $75$, and let $P$ be a Sylow-5 subgroup. Must we have $P \cong Z_5 \times Z_5$?
I think the answer is yes, based on this, but I am not supposed to use that. (I'...
- 1,867
0
votes
1
answer
136
views
group of order $p\cdot q^{2}\cdot r$, where $p,q,r$ are primes with $p<q<r$.
Is there an any reference for classifying the group of order $p\cdot q^{2}\cdot r$, where $p,q,r$ are primes with $p<q<r$?
I'm interesting in case of $p=3$, $q=5$, and $r=7$.
In M.S.E., I ...
- 1,374
1
vote
1
answer
223
views
Finite abelian group has a subgroup $H$ with cyclic quotient that "preserves" the order in the quotient.
Theorem. Let $G$ be a finite abelian group and $g \in G$. Then $G$ contains a subgroup $H$ such that $G/H$ is cyclic and the order of $gH$ in $G/H$ is the same as the order of $g$ in $G$.
Sketch of ...
- 4,125
1
vote
1
answer
95
views
Lower bounds on the number of subgroups of a specific group
I am visiting this thread:
$G$ is Abelian iff all the Sylow subgroups of $G$ are normal
I am wondering whether more can be said about the lower bounds of number of subgroups of $G$ when $G$ is of ...
- 193
4
votes
1
answer
122
views
Abelian normal subgroups of A-groups
Let $G$ be a finite solvable group, where every Sylow subgroup is abelian. I want to show that if $A\lhd G$ is an abelian normal subgroup, then
$$ A=(A\cap Z(G))(A\cap G')$$
This is easy if $A$ is a ...
- 225
0
votes
1
answer
167
views
Sylow's First Theorem acting on Abelian Group
Background
In the book of Judson's book on abstract algebra, Sylow's First Theorem is proved by first invoking the class equation and then considering the case where $p$ can/cannot divide $[G:C_G(g)]$...
- 491
0
votes
1
answer
80
views
Equivalent conditions for a Group $G$ with order $p^2q$ ( with $p>q$ both prime) be abelian.
I saw this homework many times, but always asks in the statament that $p^2 \not\equiv 1$ (mod $q$) and $q \not\equiv 1$ (mod $p$).
But today in a book text of Galois theory I Saw a similar example ...
- 1,403
2
votes
1
answer
99
views
Coprime automorphism group implies cyclic with order a cyclicity-forcing number
How do you prove Step 2 in Proof Outline, here?
- 5,409
0
votes
1
answer
942
views
Prove that there is no simple group of order $144$
I was reading the following proof for that question (Joanpemos' answer)-
How to prove a group of order $144$ is not simple using Normalizers of Sylow intersections.
And I understood it well up to ...
- 2,012
2
votes
2
answers
678
views
Group of order $7^2 \cdot11^2\cdot19$ is abelian.
I want to show any group of order $7^2 \cdot11^2\cdot19$ is Abelian, I know that $n_7=1$ and $n_{11}=1$ or $7\cdot19$ and $n_{19}=1,7\cdot11$ or $7^2\cdot11^2$. But I don't know where I should go from ...
- 970