Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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96 views

What we know about sylow subgroups of $G/P$?

Let $G$ be a finite group and $P$ be it's normal $p$-subgroup. What we can say about the sylow subgroup of $G/P$? For example if we know that a $q$-sylow $(q\neq p)$ subgroup of $G/P$ is normal and ...
3
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1answer
121 views

The structure of $(\mathbb Z/525\mathbb Z)^\times$

I am working on the following problem. Find the number of the elements of order 4 in $(\mathbb Z/525\mathbb Z)^\times$. I tried to solve it in the following way: since $525=3\cdot5^2\cdot7$, we ...
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1answer
135 views

Subgroup complement for normal subgroup of $G$ with trivial center and $\mathrm{Aut}(N)=\mathrm{Inn}(N)$

Let $G$ be a finite group and $N \subseteq G$ be a normal subgroup. If $Z(N)$ is trivial and $\operatorname{Aut} N=\operatorname{Inn} N$, then show $N$ has a complement $H$ and $H$ is normal in $G$ ...
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1answer
288 views

Existence of group of order $p$ in group of order $pq$, $p>q$

This question is related to Question on groups of order $pq$, but is different. It references the same exercise, but an earlier part. The exercise is: A group of order $pq$, $p>q$, contains a ...
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1answer
50 views

question on group representations

Here is a problem I faced in algebra. $\rho: A_4 \rightarrow End_{\mathbb C}\mathbb C^{10}$ is a representation of $A_4$. Then show that there is a vector $v \in \mathbb C^{10}$ such that $v$ is an ...
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3answers
99 views

$G$ a group of odd order. Then $\forall$ $g\in G$ there is $h\in G$ such that $g=h^2$

This one is from a practice exam I was working on. $G$ a group of odd order. Then for $\forall$ $g\in G$ there is a unique $h\in G$ such that $g=h^2$. Thoughts Well I tried a few things but they ...
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2answers
237 views

Let $G$ be a group of order $4p^n$, where $p>2$ is prime and $n>0$. Show that $G$ is not simple.

Let $G$ be a group of order $4p^n$, where $p>2$ is prime and $n>0$. Show that $G$ is not simple. (Hint: Consider the standard action of $G$ on $G/P$, where $P$ is a $p$-sylow subgroup.) Let ...
3
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2answers
192 views

The maximim number of elements in the Alternating group of degree 28

I know if $r$ is a prime number, then $(r-1)!$ is maximum number of elements of same order in the alternating group $A_{r}$. What is maximum number of elements of same order in the alternating group ...
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1answer
84 views

To classify finite $p$ group with special property

I wish to classify following finite $p$ group. Let $G$ be a finite $p$ group with the property whenever $H$ is a non normal subgroup of $G$ of order $p$, $G$ is the semidirect product of $H$ and a ...
2
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1answer
63 views

Non vanishing of group cohomology

Let $G$ be a finite group, then $H^n(G,\mathbb{Z})\neq 0$ for infinitely many $n$. This is not hard to see for cyclic groups. Can we prove this fact algebraically, could anyone provide a reference? ...
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1answer
137 views

A group of order 16 has a normal subgroup of order 4

Let $ G$ be a group of order $16$. Show that $G$ must contain a normal subgroup $H$ of order $4$. I tried the Sylow first theorem, that is $\{e\}\triangleleft H_1\triangleleft H_2\triangleleft ...
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1answer
419 views

Counter examples in group theory [duplicate]

Let $G$ be a finite group with normal subgroups $N_{1}$ and $N_{2}$. Find counter examples to the following statements 1) If $N_{1}\cong N_{2}$ then $G/N_{1}\cong G/N_{2}$ 2) If $G/N_{1}\cong ...
4
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1answer
194 views

If $x,y,z \in \mathbb{Z}$ such that $x^4+y^4+z^4 \equiv 0 \pmod{29}$, prove that $x^4+y^4+z^4 \equiv 0 \pmod{29^4}$

If $x,y,z \in \mathbb{Z}$ such that $x^4+y^4+z^4 \equiv 0 \pmod{29}$, prove that $x^4+y^4+z^4 \equiv 0 \pmod{29^4}$. I have no idea where to start, but this is my abstract algebra homework, so I ...
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1answer
37 views

Structure of stabilizer of nonsingular line in $\Omega(7,q)$

Let $q$ be a prime power, $\varepsilon=\pm$, and $M$ be the stabilizer of a nonsingular line in $\Omega(7,q)$ such that $M=\Omega^\varepsilon(6,q).2$. Then can we know more explicit structure of $M$? ...
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1answer
63 views

Definition of $\Omega$-group and $\Omega$-composition series

What are the definitions of $\Omega$-group and $\Omega$-composition series? No luck searching on the internet..
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2answers
338 views

Number of distinct groups of order n upto isomorphism, for a fixed integer n.

Given any positive integer $n$, is the number of distinct groups of order $n$ upto isomorphism finite? Thanks in advance.
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1answer
68 views

Diophantine equation and cyclicity of $\mathbb{F}_p^*$

I am trying to prove that the diophantine equation $$1998^2x^2+1997x+1995-1998x^{1998}=1998y^4+1993y^3-1991y^{1998}-2001y$$ has no solution in integers (given that $1997$ is a prime). To do so, ...
2
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1answer
273 views

On direct product of two nonabelian simple groups

I'd appreciate it if you consider this question and together with its hint: Let $G=AB$ be a finite group which is the internal direct product of $A$ and $B$ which are non-abelian simple groups. Show ...
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1answer
190 views

Existence of an element of a order 2 in the center of a finite group.

Do you have an idea how to tackle this problem: The center of a finite group G whose order is 44, has an element of order 2. I believe the idea must somehow involve the Sylow's Theorem; for example ...
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2answers
150 views

Center of a finite group of order pq

I have no clue if it is true or false: the center of a group of order pq, where p & q are distinct primes is either the trivial subgroup or the group itself! Thanks!!
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1answer
103 views

Determining the number of elements of order n in a finite group which has k cyclic subgroups of order n.

Let n,k be positive integers and let G be a group which has k cyclic subgroups of order n. Determine with proof the number of elements of order n in G. For example, a finite group G which has 28 ...
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3answers
419 views

If $G$ is a finite group, $H$ is a subgroup of $G$, and there is an element of $G/H$ of order $n$, then there is an element of $G$ of order $n$

$G$ is a finite group, $H$ is a normal subgroup. There is an element in $G/H$ of order $n$. Prove that there is an element in $G$ of order $n$. Note: I have proved that there exists an element in ...
2
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1answer
96 views

Fiber Product of finite perfect groups

Is the class of finite perfect groups closed under fiber products? In other words if $G$ is a finite group, $N_1$, $N_2$ normal subgroups with trivial intersection such that $G/N_1$ and $G/N_2$ are ...
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0answers
232 views

Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal.

Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal. Let $P_k$ denote the $k$-Sylow subgroup and let $n_3$ denote the number of conjugates of $P_k$. $n_2 \equiv 1 ...
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2answers
135 views

Order of elements in $S_4$

Let $r(n) = \left| \left\{ \sigma \in S_4 : \mbox{ord} ( \sigma) = n \right\} \right|$. Is it true that: $r(2)>r(4)$ $r(4) > r(3)$ $r(1)+r(3) = r(2)$ $r(5) = r(6)$ I can write all elements ...
5
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3answers
165 views

Prove that $D_3 \oplus D_4$ is not isomorphic to $D_{12}\oplus\mathbb Z_2$

This is a problem from the textbook, doing this for practice and not assignment. Prove that $D_3 \oplus D_4$ is not isomorphic to $D_{12}\oplus\mathbb Z_2$. So we know $|D_3| = 6$ and $|D_4| ...
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1answer
99 views

Questions about group of automorphisms

Let $G$ be a finite $p$-group. Suppose that $H,K$ are subgroups of $G$, such that $H \leq K$. If $\operatorname{Aut}(H)$ denotes the group of automorphism of $H$, then is there any relationship ...
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3answers
56 views

Subgroup Question

Let $G$ be a group and let $A \subset G$. I want to show that there is a $\subset$-least subgroup $H$ of $G$ such that $A \subset H$. The subgroup is noted as $\langle A\rangle$. Then I want to show ...
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1answer
87 views

Why are regular $p$-groups called “regular?”

In the concept of regular $p$-Groups, what does "regularity" stand for? What is "regular" in such groups? I would like to know idea behind defining these groups, and naming these groups "regular." ...
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0answers
92 views

What is the smallest $n$ for which the usual “counting sizes of conjugacy classes” proof of simplicity fails for $A_n$?

A common proof of the simplicity of $A_5$ proceeds as follows. First, note that a normal subgroup is always a union of conjugacy classes. Also, a subgroup has order dividing the size of the group by ...
0
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1answer
71 views

How to prove that if $p^k$ divides a finite group $G$ there exists a strict subgroup $p^k||H|$ or $p|Z(G)$

Given a finite group $G$ and a prime $p$ such that $p^k| |G|$. Now prove that there exists a strict subgroup $H$ of $G$ such that $p^k | |H| $ or $p| |Z(G)|$. Well, I know what you're thinking: just ...
3
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1answer
60 views

Order of some subgroup of $\mathit{SL}_2(\mathbb Z/p^n\mathbb Z)$

I am trying to this problem from a past exam: Let $p$ be an odd prime and $n$ a natural number. Show that the group $G:= \left\{\begin{pmatrix}a& b\\0 & d\end{pmatrix}\right\} < ...
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2answers
89 views

If $G$ is a group, is there any method which would enable us to know if G is congruent to A/B for some group A and normal subgroup B?

If $G$ is a group, is there any method which would enable us to know if $G \cong A/B$ for some group $A$ and normal subgroup $B\,?$ If there is no general method for that, in which cases can we know ...
5
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2answers
43 views

Extracting Information from Lists in GAP

If I have a list $L$ in GAP, and a certain list of properties $a,b,c$, how can I count the the number of elements in my list that have all three properties? I've searched the manual (chapter on ...
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2answers
98 views

Given a group $G$ of order $102=(2 \cdot 3 \cdot 17) $ and $|Z(G)|=2$, show that $G/Z(G)$ has a subgroup of order 17.

I can't find the answer to this question. Given a group $G$ of order $102=(2 \cdot 3 \cdot 17) $ and $|Z(G)|=2$, show that $G/Z(G)$ has a subgroup of order 17. These were my thoughts: The order of ...
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2answers
97 views

A finite group is a product of cyclic groups?

Let $G$ be a finite group and $H$ a subgroup of $G$. There exists a sequence of cyclic subgroups $H_1, ..., H_n$, not necessarily distinct, such that $H$ is the product $H = H_1H_2\cdots H_n = \{ ...
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1answer
255 views

Computing Conjugacy Classes of Subgroups in GAP

GAP has the command ConjugacyClassesSubgroups which gives a list of the conjugacy classes of a finite group $G$. Is there a way I can specify further what types of subgroups GAP reports? For ...
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2answers
122 views

Why is $\frac{G}{H \cap K}$ not cyclic if G is finite and H and K are 2 different subgroups of index 2?

I was making some exercises on group theory when I came across this problem: Suppose G is a finite group and K and H are 2 different subgroups of index 2. Now first prove $(H \cap K) \triangleleft G$ ...
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1answer
75 views

How to change a matrix to a group [closed]

a := [1,2,3;4,5,6;7,8,9]; how to change a matrix to be a group ? is representation = this general matrix or representation is another kind of matrix?
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If $N \unlhd G$, $G/N$ solvable, $\chi \in Irr(G)$, $\psi \in Irr(N)$ and $<\chi_N , \psi>_{_N} \neq 0$, then $(\chi(1)/\psi(1)) | [G:N]$.

If $N \unlhd G$, $G/N$ solvable, $\chi \in Irr(G)$, $\psi \in Irr(N)$ and $\langle\chi_N , \psi\rangle_{_N} \neq 0$, then $(\chi(1)/\psi(1)) | [G:N]$. Can anybody help me to prove that?
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2answers
96 views

Conjugacy Class of symmetry group $S_{10}$

Let $X=\{a\in S_{10} | ~~\text{order}(a)=8\}$. Determine how many conjugacy classes are in $X$. How to do this question in general?
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2answers
248 views

What is the quotient of a cyclic group of order $n$ by a cyclic subgroup of order $m$?

Suppose that $H$ is a (cyclic) subgroup of order $m$ of a cyclic group $G$ of order $n$. What is $G/H$? It's a very simple question but I am still struggling with getting accustomed to the ...
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2answers
487 views

Elementary divisors of an abelian group

From Advanced Modern Algebra (Rotman): Proposition 4.10 If $G$ is an abelian group and $p$ is prime, then $G/pG$ is a vector space over $\Bbb{F}_p.$ Definition If $p$ is prime and $G$ is a ...
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1answer
111 views

Number of Inequivalent Difference Sets In Elementary Abelian 2-groups

I have reason to believe that there is only one$(2^{2s+2},2^{2s+1}-2^s,2^{2s}-2^s)$- difference set (based on experimentation in GAP), up to equivalence/complementation, in any elementary 2-group of ...
6
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3answers
85 views

Let $G$ be a finite p-primary abelian group. If a is an element of largest order in G, then $A= \langle a \rangle$ is a direct summand of G.

I was trying to read the proof from Advanced Modern Algebra (Rotman), but there was something that seemed confusing to me. It's only the last part that's confusing, but I put the whole proof anyway. ...
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1answer
122 views

group with a non cyclic proper subgroup

I need to know which of the following group has a proper subgroup which is NOT cyclic $1. \mathbb{Z}_{15}\times \mathbb{Z}_{77}$ $2. S_3$ $3. (\mathbb{Z},+)$ $4.(\mathbb{Q},+)$ any finitely ...
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2answers
83 views

Questions about factor groups of $\mathbb Z$

I'm not familiar with modules so much and my teacher didn't go over it very well. So I was wondering if someone could help on some of these...at least a starting point. List the elements of $A \in ...
2
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0answers
76 views

Partition of Symmetric Group

For symmetric group $S_n$, we need to find a collection of subgroups $G_i$'s such that union of these subgroups is the group $S_n$ and each subgroup found is isomorphic to direct product of cyclic ...
6
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1answer
286 views

Finite group with elements of given order

A few weeks ago, there was a queston on MSE that got edited, as soon as the question was answered well enough, according to OP. In the end, I think this question got reversed to its original question ...
1
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1answer
79 views

non-split extension of the simple group $L_3(4)$

I would like to know the structure of the groups $L_3(4).C_2$ and $L_3(4).C_{11}$. (By $C_n$ I mean the cyclic group of order $n$ and by $G=K.L$ I mean the non-spli extension of $K$ by $L$, were $K$ ...