All Questions
60 questions
0
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73
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non-normal subgroups of a finite $p$-group
throughout studying a paper about finite $p$-groups,
I have a question as follows,
Let $G$ be a finite non-abelian $p$-group, where $p$ is odd and $Z(G)$ is cyclic.
($Z(G)$ denotes the center of $G$). ...
- 655
0
votes
1
answer
84
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How to find subgroups of direct product of cyclic groups [duplicate]
Find all subgroups of $\mathbb{Z_2} \times \mathbb{Z_5} \times \mathbb{Z_7}$.
I know that this would be isomorphic to $\mathbb{Z_{70}}$, so that I would be looking for $8$ subgroups because $70$ has $...
- 83
5
votes
4
answers
334
views
Show that a subgroup of order $3$ is normal in a group of order $15$
I've been trying to show that any group of order $15$ is cyclic and the only missing part in my proof is to show that the subgroup with $3$ elements (which we know exists by Cauchy's theorem) is ...
- 520
0
votes
1
answer
169
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Sylow $p$-subgroups are all cyclic then $G$ has normal subgroup $N$ and such that $G/N$ and $N$ are both cyclic
Problem: If $G$ is a finite group whose Sylow $p$-subgroups are all cyclic then $G$ has normal subgroup $N$ and such that $G/N$ and $N$ are both cyclic.
Whenever I need to find normal subgroup, I ...
- 721
0
votes
0
answers
58
views
A depict of a cyclic group with a normal group
If $G$ is a group, $H$ is a normal subgroup of $G$, $[G:H]=k,{\rm ord}_G(a)=n,a\in G,|\langle a\rangle\cap H|=m,$ prove: The term of $\langle a\rangle$ in $G/H$ is $\frac{n}{m}$.
I have some thoughts ...
- 57
2
votes
1
answer
121
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Is the number of finite subgroups of a finite-by-cyclic group finite?
Let $G$ be a finite-by-cyclic group, i.e. it contains a finite normal subgroup $N$ such that $G/N$ is cyclic.
My question is that: Does $G$ contain only finitely many finite subgroups?
What I've tried:...
- 2,803
2
votes
2
answers
156
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Does a finite-by-(infinite dihedral) group have the form $N\rtimes H$ for a finite normal subgroup $N$ and a cyclic subgroup $H$?
Let $G$ be a finite-by-(infinite dihedral) group, i.e. it contains a finite normal subgroup $K$ with $G/K$ isomorphic to the infinite dihedral group.
Question: If $G$ is a finite-by-(infinite dihedral)...
- 2,803
2
votes
2
answers
70
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Show that there exist a cyclic normal subgroup $N$ of $G$ whose order is $255$ such that $G/N$ is cyclic
I was practicing for my Group Theory exam and I could not solve the following question:
Let $G$ be a group of order $255=3\cdot 5\cdot 17$. Find a cyclic normal subgroup $N$ of $G$ such that $G/N$ is ...
0
votes
1
answer
94
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A problem in group theory and representation theory.
Suppose that $H = \{1, h, h^2, \ldots, h^{n-1}\}$ is a normal subgroup of a finite non-abelian group $G$
having order $n$. It is known that $H$ is cyclic with generator $h$. Let $c_G$ be the number ...
- 4,592
-1
votes
1
answer
47
views
Is every normal subgroup of some group also a cyclic subgroup? [closed]
I know the converse doesn't hold, but I am unable to find a counterexample. Yet I don't know how normal subgroups bring rise to a generator.
0
votes
1
answer
402
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Question about "Quotient Group of Cyclic Group is Cyclic"
I found a proof of the fact that
if $G$ is a cyclic group and $H$ is a subgroup of $G$, then $G/H$ is a cyclic subgroup.
They don't mention that $H$ is a normal subgroup. But to define the quotient ...
0
votes
0
answers
69
views
Show by giving a counterexample that $H \cap N$ is not necessarily a normal subgroup of $G$.
Let $G$ be a group and $H$ a subgroup of $G$ and $N$ a normal subgroup of $G$. Show that $H \cap N$ is a normal subgroup of $H$ and show by giving a counterexample that $H \cap N$ is not necessarily a ...
- 185
0
votes
1
answer
124
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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 ...
2
votes
2
answers
61
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Find the order $\bar{5} + H$
Let $G=\mathbb{Z}/18\mathbb{Z}$ and $H=\langle 6+ 18\mathbb{Z}\rangle= \langle \bar{6}\rangle $ a subgroup of $G$. Consider the group $G/H$. What is the order of $(5 + 18 \mathbb{Z}) + H \in G/H$?
...
user994888
1
vote
3
answers
398
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Show that dihedral group of twice odd order doesn't have a normal subgroup of order $2m$, where $m$ divides $n$.
Let $n \ge 3$. Let $D_n = \langle r,s \rangle$, for $r^n=s^2=1$ and $rs=sr^{-1}$. Then $D_n$ (not $D_{2n}$) is the dihedral group of order $2n$.
Show that if $n$ is odd, then $D_n$ doesn't have a (...
- 14k
4
votes
1
answer
124
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For a group $G$, $N\unlhd G$, $G/N\cong\Bbb{Z}/a\Bbb{Z}$ and $N\cong\Bbb{Z}/b\Bbb{Z}$, where $b<a$ and $(b,a)=1$, show $G$ is abelian.
I've been pounding my head on my desk attempting to figure out how to start this proof:
If we have a group $G$, a normal subgroup $N\trianglelefteq G$, $G/N \cong \mathbb{Z}/a\mathbb{Z}$, and $N \...
- 195
0
votes
0
answers
92
views
Group of order 35 must have a normal subgroup of order 5 or 7 or both without using Sylow’s theorems [duplicate]
This was a previously asked question that only had answers referring to Sylow's theorem, which is not something I can use here.
Let $G$ be a group, $|G|=35.$ We know that $G$ must contain an element ...
- 724
1
vote
1
answer
157
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For $|G|=pq$ ($p$ & $q$ are distinct primes), $H$ and $K$ are normal subgroups of order $p$ and $q$ resp., then, prove that $G$ is cyclic
My Attempt:
Since $H$ and $K$ are groups of prime order, so, $H$ and $K$ are cyclic.
Let $H=\langle a\rangle $ and $K=\langle b\rangle $, for some $a$,$b$ in G. Then $o(a)=p$, $o(b)=q$
Now we consider ...
- 716
1
vote
0
answers
33
views
Efficient way of writing the elements of a quotient group. [duplicate]
I am considering the normal subgroup $H=\langle[28]\rangle$ of the additive group $\mathbb{Z}_{220}$. I want to obtain the elements of the quotient group $\mathbb{Z_{220}}/H$.
Since $H=\langle14[2]\...
- 904
0
votes
3
answers
639
views
Prove or disprove: If $H$ is a normal subgroup and cyclic and $G/H$ are cyclic, then $G$ is cyclic.
Prove or disprove: If $H$ is a normal subgroup and cyclic and $G/H$ are cyclic, then $G$ is cyclic.
I don't understand the quotient group $G/H$ being cyclic. What does it mean? From what I understand,...
- 3
4
votes
1
answer
348
views
Cyclic normal subgroup of perfect group is in the center
I've been trying for a while to solve an exercise/prove a proposition, which at first seemed elementary, but now I even doubt if it's a true proposition. The proposition is:
Let $G$ be a perfect group ...
0
votes
1
answer
53
views
Let $G$ be a group. Let $N\unlhd G$ with $G/N$ being cyclic. Let $H\leq G$. Show that $HN\unlhd G$. [closed]
Let $G$ be a group. Let $N\unlhd G$ with $G/N$ being cyclic. Let $H\leq G$. Show that $HN\unlhd G$.
I cannot figure out how to use the fact that $G/N$ is cyclic in this proof. Any help is much ...
- 647
0
votes
1
answer
235
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Prove that $G$ is cyclic group of prime order [duplicate]
I'm at the beggining of a Group Theory course, and I'm trying to solve this problem:
Let $G = 1$ (being $1=\{e\}$, $e$ the neutral element in $G$) be a group containing no subgroup different from $1$ ...
- 3,609
2
votes
1
answer
326
views
Checking whether the group generated by a set is normal
As part of a homework problem, we proved that we can test whether an element $g$ of a finite group $G$ is in the normalizer of a cyclic subgroup $H=\langle x\rangle$ by conjugating just the generator $...
- 109
1
vote
0
answers
55
views
Are infinite cyclic normal subgroups "virtually" central?
Let $G$ be a finitely generated group and let $C$ be an infinite cyclic normal subgroup of $G$. My question is: Does $G$ necessarily have a finite-index subgroup $H$ such that $C \le Z(H)$? If not, ...
- 429
1
vote
2
answers
51
views
Using Definition of Cyclic Group to prove B is a Subgroup
Given the Dihedral group $ D_4 $ (that is where $ D_4 = $ { $ id, R, R^{2}, R^{3}, F, RF, R^{2}F, R^{3}F $} ); Let $B =$ {$id, RF$}
I now wish to prove that $B$ is a subgroup of $D_4$:
Note that $B =...
2
votes
2
answers
242
views
Preimage of cyclic subgroup under projection is Abelian
Let $G$ be a finite non-Abelian group, let $Z(G)$ denote its centre, let $\pi: G \to G/Z(G)$ be the canonical projection, and let $H \vartriangleleft G/Z(G)$ be a cyclic normal subgroup of the ...
- 2,907
0
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1
answer
219
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When is a quotient group of a free group finite?
Let $I=\{\alpha\}$ and $k\in\mathbb{N}$. Consider the free group $F(I)$ constructed on $\{\alpha\}$. Let $\phi_\alpha$ be the canonical homomorphism of $\mathbb{Z}$ into $F(I)$. Let $r=\phi_\alpha(1)^...
0
votes
5
answers
602
views
Can $G/H$ be cyclic if $G$ is nonabelian?
I've learned in class that if $G/Z(G)$ is cyclic then $G$ is abelian.
I was wondering if $H\lhd G$, where $G$ is nonabelian if $G/H$ is cyclic, when $H$ is not $G$ itself. If so, could someone give ...
- 327
-1
votes
2
answers
630
views
Assume $N \triangleleft G$ Prove that if $[G:H]$ is a prime, then $G/N$ is cyclic. prove or disprove the converse of the first statement.
Assume $N \triangleleft G$ Prove that if $[G:N]$ is a prime, then $G/N$ is cyclic. prove or disprove the converse of the first statement.
Doesn't LaGrange state that if the group is prime, then it ...
- 2,325
2
votes
1
answer
93
views
Locally cyclic quotient and locally cyclic subgroup
Suppose $G$ is a group, and there exists a finite normal subgroup $H$ of $G$, such that $\frac{G}{H}$ is locally cyclic. Is it always true, that in this case $G$ has a locally cyclic subgroup of ...
- 15.8k
-1
votes
1
answer
309
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normalizer of a cyclic group
I have a cyclic permutation group G=<(123456)> what is normalizer of this group?
the normalizer a subset A of a group G such that:
$$N_{G}(A)= (g \in G s.t \;\;\; gag^{-1} \in A , \;\; \forall a \...
- 115
4
votes
1
answer
251
views
Free normal subgroup of an HNN-extension
Suppose $F$ is a finitely generated free group and $a,b$ are not in $F'$ but $b^{-1}a \in F'$. By taking the HNN extension $G=\langle F,t | t^{-1}atb^{-1}\rangle$, is there a way to find a normal free ...
- 1,189
3
votes
3
answers
87
views
Show that $G$ = $(\mathbb Z_3 × \mathbb Z_4, +)$ is cyclic
Let $G$ = $(\mathbb Z_3 × \mathbb Z_4, +)$ with the operation defined somewhat like vector
addition: for $a, a'$ ∈ $\mathbb Z_3$ and $b, b'$ ∈ $\mathbb Z_4$,
$(a, b) + (a', b') = (a +_3 a', b +...
- 1,090
9
votes
1
answer
2k
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Find the normalizer of a cyclic subgroup of $S_7$
Let $P\subset S_7$ be a cyclic subgroup of order $7$. Show that the normalizer $N$ of $P$ has order $42$, and find a pair of cycles generating $N$.
My attempt
First note that the cyclic subgroup is ...
- 633
2
votes
0
answers
57
views
Are two lifts of a generator of a cyclic quotient conjugated?
Let $G$ be a group and $I$ be a normal subgroup of $G$ such that $G/I$ is cyclic, and let $\bar{g}$ be a generator. Let $g, g' \in G$ be two lifts of $\bar{g}$, i.e. $g + I = g' + I = \bar{g}$.
...
- 3,185
0
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1
answer
937
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In proving G contains an element of order 15 if contains normal subgroups of orders 3 and 5, respectively, is $HK$ itself cyclic with order 15?
There is an answer here, but it is a "roadmap". group containing normal subgroups of orders $3$ and $5$ contains element of order $15$ There are answers here, but they are "roadmaps" too. If $G$ ...
user198044
3
votes
1
answer
93
views
$H\lhd G$ implies $C_G(h)/C_H(h)$ is cyclic
The actual question is as follows:
Let $G$ be a finite group of odd order, $H$ a subgroup of $G$ of index 3. Let $h\in H$. Prove that $C_H(h) \lhd C_G(h)$ and $C_G(h)/C_H(h)$ is cyclic (where $C_G(h)...
- 554
2
votes
1
answer
109
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Group theory, subgroup of index 2
I have a group $G $ which is an abelian group such that $\mathbb R $ is a subgroup of index $2$. Can I conclude that $G $ is isomorphic to $\mathbb Z /2\mathbb Z \times \mathbb R $ ?
I am not sure, ...
- 1,322
3
votes
1
answer
61
views
If a certain factor group is cyclic, can we transfer this to another factor group?
Let $G$ be a finite non-abelian group with cyclic normal subgroup $N \lhd G$ such that $G/N$ is also cyclic.
Let $H \lhd G$ be another cyclic normal subgroup of $G$ with $H \leq N$.
We know $G/N \...
- 508
0
votes
1
answer
715
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is the group cyclic, if Normal Subgroup and Quotient group are cyclic. [duplicate]
Prove or disprove that if $H$ is a normal subgroup of a group $G$ such that $H$ and $\frac{G}{H}$ are cyclic, then $G$ is cyclic.
I am not sure if the above question has any mistake.
I know ...
- 513
6
votes
1
answer
8k
views
Is it true that cyclic subgroups are always normal?
There is a proposition that I'm supposed to "prove", but it doesn't sound true to me. It says that if $H$ is a cyclic subgroup of a group $G$ (notation $H<G$), then every $K <H$ is normal in $G$....
- 776
0
votes
1
answer
205
views
Prime index normal subgroup? [closed]
When it comes to studying normal subgroups, how would one go about proving the following?
Prove that if $N\triangleleft G$ and $G/N$ is cyclic, then there exists a subgroup $N'\triangleleft G$ such ...
- 1,918
0
votes
1
answer
5k
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If every cyclic subgroup of a group G be normal in G, prove that every subgroup of G is normal in G. [duplicate]
If every cyclic subgroup of a group G be normal in G, prove that every subgroup of G is normal in G.
Attepmt
Let G be a group. Let H be a Normal subgroup of G. Let $K=\langle a \rangle$ be a cyclic ...
- 4,080
-1
votes
2
answers
389
views
Prove or disprove if G is an abelian and H is a normal subgroup of G,then H must be a cyclic
$G$ is an abelian group,so every subgroup of an abelian group is normal and:
$$o(H)|o(G)$$
$G$ is an abelian group:$$\forall a,b:ab=ba$$
$H$ is a normal subgroup:$$\forall g \in G,\forall h \in H: ghg^...
- 131
0
votes
3
answers
171
views
Finding the Order of G, given subgroups
I need help with a practice exam question:
$G$ is a cyclic group that has only 3 subgroups: $e$ (identity), $G$, and $G'$ s.t. |$G'$| = $n$. What is |G| if:
a) $n = 5$
b) $n$ is any prime number $p$...
- 1,660
3
votes
4
answers
4k
views
Show that $\mathbb{Z}$ has no composition series.
Please, read the whole post before trying to answer.
Remark: here $\subset$ means "strict inclusion".
I need to prove that the group $\mathbb{Z}$ has no composition series. That is, no ...
- 4,648
2
votes
2
answers
126
views
Quotient of $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$ by $\langle a^2 \rangle$
Let $G$ be finite group of order $8$ of the form: $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$.
The elements are $\left\lbrace 1, a, a^2, a^3, b, ab, a^2b, a^3b\right\rbrace$....
- 2,199
6
votes
5
answers
3k
views
Nature of $G$ when $N$ is cyclic, normal subgroup of $G$ and $G/N$ is cyclic
Let $N$ be a normal subgroup of $G$ and both groups $N$ and $G/N$ are cyclic. I need to prove that $G$ is generated by at most two elements.
To that effect, what sorts of things do we know about $G$ ...
user100463
0
votes
1
answer
64
views
Does $H \unlhd G$ imply $G/H$ is cyclic and therefore abelian?
I have a review question which wants me to show equivalency between the two ends of the question, and I came to the conclusion one way top show it is abelian would be to prove it is a cyclic group, ...
- 312