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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1answer
219 views

A Finite Group of Nonprime Order which is Unique up to Cyclic Group

Maybe I have to wait until I learn and study more, but I just became curious. I know that every finite group of prime order is cyclic, and hence unique up to isomorphism. I have 2 questions about this....
3
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1answer
124 views

Factorizing elements of a group into a product of generators.

$$ s = (1\ 2) \\ t = (1\ 2\ 3\ ...\ n) $$ Given the Symmetric Group $S_n$ generated by $s$ and $t$, is there a way to quickly factor an element $g \in S_n$ into a minimal product of positive powers ...
3
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2answers
45 views

Prove that $S= \langle(1,2,3,4) \rangle$ has 3 conjugates in $S_4$

Some things I know: $S = \{ (1),(1,3)(2,4), (1,2,3,4),(1,4,3,2)\}$ $(2,4) \in N_G(S)$ Number of conjugates = $[G: N_G(S)]$ This seems like such a easy question but it made me realised that I do ...
2
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1answer
50 views

Question on simple subgroup $H$ and a normal subgroup $N$, of $G$

This one is a bit strange to me, mainly the third hypothesis. It goes as follows: Given a group (finite) group $G$, and $N, H \leq G$ such that $N$ is normal in $G$, and $H$ is simple $...
3
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3answers
122 views

Example where a finite group $G$ of order $n$ has no subgroup of order $m$

Using the Fundamental Theorem of Abelian Groups, one can prove that if $G$ is a finite abelian group of order $n$ such that $m$ is a positive integer that divides $n$, then $G$ contains a subgroup of ...
3
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1answer
126 views

Finite group with unique subgroup of each order. [duplicate]

Let $G$ finite group, and suppose $G$ has unique subgroup of each order (which divides $G$'s order) - Show that $G$ is cyclic. I reduced the problem to sylow subgroups of $G$ (they are all normal), ...
2
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1answer
85 views

How unitarize an irreducible representation of a finite group?

Let $G$ be a finite group acting irreducibly on the space $V$. $$\psi : G \to Aut(V)$$ Question: How unitarize the representation $V$? I'm looking for a computable process.
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3answers
61 views

Is the map $f:S_n \to A_{n+2}$ a homomorphism where $f(s)=s$ when $s$ is even and $f(s)=s \circ (n+1,\ n+2)$ when $s$ is odd?

Is the map $f:S_n \to A_{n+2}$ given by $$f(s)= \begin{cases} s & s\ \text{is even}\\ s \circ (n+1,\ n+2) & s\ \text{is odd} \end{cases}$$ an injective homomorphism? I can show that if it ...
3
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1answer
139 views

Given a finite set, how to generate all possible groups defined on it?

Just started learning algebra, the "group" concept looks simple but more thoughts are needed. Given a finite set $S$, say, with $n$ elements, how can we generate all possible groups on $S$? Is there ...
2
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1answer
88 views

Proving that a subgroup $|H|=p^k$ is a Sylow subgroup of $|G|=p^km$, $m\nmid p$

I'm attempting to prove Sylow's theorems following the sketch described in the Wikipedia article, but I've run into a little hitch since the theorems are presented in a few slightly different forms in ...
3
votes
1answer
93 views

let $x$ be in finite group $G$ and let order of $x$ is $p$. If $h^{-1}xh = x^{10}$ for a finite group show that $p=3$ [duplicate]

Let $G$ be a finite group, $p$ be the smallest prime divisor of $|G|$ and x $\in$ G an element of order $p$. Suppose $ h \in G $ is such that $h^{-1}xh = x^{10}$. Show that $p = 3$. I cant ...
1
vote
1answer
75 views

Discrete quotient group

I have a hard time understanding quotient groups. For example, I need to make sense of the expression $$\mathcal{S}_3 (1,3,5) / \mathcal{Z}_2 (3,5).$$ Here, $\mathcal{S}_3 (1,3,5)$ is a symmetric ...
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0answers
36 views

Adding relators and normal closure

I learned the notion of group presentation. By definition, a group described by a presentation is the quotient of some free group by the normal closure of the relators. In general, given a group ...
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0answers
103 views

What does $aba^{-1}b^{-1} \notin H$ imply?

​I am working on a problem on commutator subgroup of finite group. Long story short, I was given $H < G$ and $H' \neq H$ and am aiming to prove $H \lhd G$. As you know that $H'$ is commutator ...
3
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1answer
81 views

on the classification of groups of order $p^4$.

Burnside, in his book "Theory of Groups of Finite Order" (see http://www.gutenberg.org/ebooks/40395) classify all the groups of order $p^4$ (see pages 100-102). My question is in regard to the group (...
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2answers
88 views

Splitting of conjugacy class in $A_n$

During reading, I have encountered this, in several places: The following are equivalent for a permutation $\sigma \in A_n$: 1) the $S_n$-conjugacy class of $\sigma$ splits into two $A_n$-classes 2) ...
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1answer
181 views

How getting the unitarized irreducible representations with GAP?

The function IrreducibleRepresentations on GAP gives non-necessarily unitary representations, for example: ...
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1answer
67 views

Operation with normal subgroup

I am working on a problem on finite group theory, and would like asking a question on the correct operation of normal subgroup. Suppose that $H$ is normal subgroup of $G$ and the factor group $G/H$ ...
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0answers
17 views

If G is a p-group, the number of nonnormal subgroup is a multiple of p? [duplicate]

I want to show that If $G$ is a finite $p$-group, then the number of nonnormal subgroups of $G$ is a multiple of $p$. I think I need to consider a conjugation action. But then?
1
vote
1answer
127 views

Find number of Sylow $3$ and Sylow $5$ subgroups of a simple Group, $G$ of order $60$

So I wanted to check if what I did was correct. I'm not sure if it is and if so what would the correct way to go about this be? So firstly $|G| = 2^2*3*5$. This confirms there is Sylow 3 - Subgroup'...
2
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2answers
97 views

Kernel of a homomorphism of a group algebra

Excuse me, this is stupid, but I have a short circuit in my head, I can't understand the situation. Let $k$ be a field, $G$ a finite group, $kG$ the corresponding group algebra and $\delta:G\to kG$ ...
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2answers
84 views

To make $(K_4,+)$ ( the Klein-4 group ) a ring

How can we define an operation $.$ such that the Klein-4 group $(K_4,+)$ becomes a ring $(K_4,+,.)$ ?
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1answer
70 views

Ways of writing an $n$-cycle as product of a $2$-cycle and $n-1$ cycle.

We know that any $n$ cycle can be written as a product of a $2$-cycle and an $n-1$ cycle; but this decomposition is not unique: $(123)=(12)(23)$ and $(123)=(23)(31)$ [product taken from right to left ...
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1answer
78 views

A sub-theorem inside “Prove that The group $A_4$ has no subgroup of order $6$”.

Proposition: The group $A_4$ has no subgroup of order $6$. Proof: Suppose we have some $H$ such that $|H|=6$, thus $[A_4 : H ] = 2$. Thus there are only two cosets of $H$ in $A_4$ . Inasmuch as one ...
2
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0answers
51 views

Sylow 5-subgroups of groups of order $2^n5^m$ are normal

My textbook says: Show that a group of order $2^n5^m, m, n \ge 1$ has a normal 5-Sylow subgroup. I've been banging my head against this problem for days with no success, how can I prove this?
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2answers
54 views

Number of Sylow $2$-subgroups in dihedral group $D_{20}$

By Sylow's theorem I know that the number of Sylow $2$-subgroups in the symmetry group of a regular $10$-gon $D_{20}$ is either $1$ or $5$. How do I exclude the possibility $1$?
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0answers
150 views

Why do Sylow $p$-groups in finite simple group have trivial intersection?

I'm looking for an argument for the fact that two Sylow $p$-subgroups in a finite simple group have trivial intersection. In the case that the order of the Sylow subgroups is the prime $p$ it is easy, ...
2
votes
1answer
83 views

What is set builder of $\langle H, K \rangle$?

I am looking left and right for a lemma to solve a problem on solvable group here, and I think I have found one under commutator group: For any two subgroups $H, K$ of $G$, the $[H, K]$ is a ...
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1answer
78 views

What is the composition series of $\mathbb Z_7$ x $\mathbb Z_{12}$

So I get the answer as follows (which is correct I believe): {$0$} x {$0$} $\vartriangleleft$ $\mathbb Z_7$ x {$0$} $\vartriangleleft$ $\mathbb Z_7$ x <$4$> $\vartriangleleft$ $\mathbb Z_7$ x &...
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0answers
33 views

group cohomology of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $F$ be a field. What is the cohomology $$ H^*(\Sigma_k;F)=H^*(K(\Sigma_k,1);F)=H^*(B\Sigma_k;F)? $$ For $F=\mathbb{Z}/p\mathbb{Z}$ for prime $...
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2answers
174 views

Reconciling Different Definitions of Solvable Group

Looks like my class note defines solvable group differently from others: A finite group $G$ is called solvable if $H' \neq H$ for each subgroup $H$ of $G$ different from $\{1\}$, where $H'$ is ...
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1answer
62 views

Basic group algebra exercise

I'm working on this abstract algebra problem that was given as homework to me, I'm in my first year of university and have a semester's background in basic abstract algebra. Here is the problem: Let $...
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0answers
77 views

Kernel and image of a homomorphism $SL(2,5)\to S_5$

Since $SL(2,5)$ has a subgroup of index $5$, I can use the left coset action to define a homomorphism between $SL(2,5)$ and $S_5$. How can I find the kernel and the image of this homomorphism? ...
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0answers
132 views

Capable group of order 32

A group that can be written as $G/Z(G)$ for some group $G$ is called capable. Can someone list the capable groups of order 32?
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2answers
251 views

Elementary Applications of Cayley's Theorem in Group Theory

The Cayley's theorem says that every group $G$ is a subgroup of some symmetric group. More precisely, if $G$ is a group of order $n$, then $G$ is a subgroup of $S_n$. In the course on group theory, ...
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3answers
192 views

Group acting on its subsets

Let $ G $ be a group with $ |G|=mp^\alpha $ where $ \alpha\geq1 $ and p is prime integer with $p \nmid m$. Then denote the set of subsets of G, having $p^\alpha$ size, with $X$. Then with the action $...
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3answers
81 views

Showing that $\{0, 1\}$ is a group under addition modulo $2$

I'm considering a set $ G = \{0,1\}$ under addition modulo 2. I.e. $$ a*b = a + b\bmod 2, \quad \quad \forall \ a,b \in G. $$ I am able to show that there exists an identity element, $0$. Showing ...
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0answers
61 views

on the characters of the normal subgroup and its quotient

I read character theory recently and thought about the following proposition, but I do not know is this true or false: Let $G $ be a finite group such that two distinct primes $ p $ and $ q $ ...
0
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1answer
78 views

Finite Group is Subgroup of Its Radical's Automorphism

I am still working on this problem on radical of finite group: Assume that $R(G)$ is simple and not commutative, show that $G$ is a subgroup of $Aut(R(G))$. I have managed to parse the problem ...
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1answer
38 views

Find a composition series of $D_8$

Answer: Let $D_8=\langle a,b\rangle=\{e,a,a^2,a^3,b,ba,ba^2,ba^3\}$, where $a^4 = b^2 = e$ and $ab = ba^{-1}$ as usual. Then $$\{e\}\lhd\langle a^2\rangle\lhd\langle a\rangle\lhd D_8$$ or $$\{e\}\...
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2answers
210 views

No group of order $400$ is simple - clarification

I was reading through a proof that no group of order $400$ is simple which can be found here: http://math.stackexchange.com/a/79644/169389 Here is an outline for a solution. First of all, $|...
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64 views

Reference request: groups of order $p^4$.

I am looking for a textbook or a paper which include the classification of groups of order $p^4$ ($p$ is prime) using generators and relations. In particular I like to understand which group $G$ "...
4
votes
1answer
140 views

Calculating the second cohomology group for trivial group action

Let $G$ be a finite group acting trivially on $\mathbb{R}^*$. How can I compute $H^2(G,\mathbb{R}^*)$? It seems that direct calculations are somewhat hopeless, but the answer should be simple anyway.
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1answer
82 views

Simple Groups of Finite Order

Prove that there is NO simple group of order $n$ for each the following integers: $n=88, n=96, n=132.$ I am supposed to solve this using Sylows theorems somehow. Lets start with $n = 88$ and say we ...
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1answer
113 views

What is a generator of a finite cyclic group? (General)

I have asked a few questions about this but I am still confused. So, in general, what is a generator of a finite cyclic group and how is it found? I have seen in books and my notes a lot of ...
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0answers
75 views

A group of order $2^67$

Let $G$ be a finite group of order $2^6\times 7$. Also $G$ has at least $5$ irreducible characters of degree $7$. I want to prove that with this condition there exists no character of even degree in $...
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1answer
126 views

Proof-verification group order 48 not simple

Is the following proof correct? Let $G$ be a group of order 48. Let's prove it is not simple. $\lvert G\rvert=48=2^4\cdot3$. By Sylow's Theorem, $n_3\in\{1,4,16\},\:n_2\in\{1,3\}$. $n_3=1$ or $n_2=...
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0answers
62 views

$\lvert G\rvert=24$ not simple by a counting arguments [duplicate]

I want to prove that $\lvert G\rvert=24$, then it has a non-trivial normal subgroup. Here is my attempt: $n_2$: number of 2-Sylow subgroup; $n_3$: number of 3-Sylow subgroup. $n_2\in\{1,3\},\:\:n_3\...
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1answer
155 views

$|G|=24$ prove that $G$ is not simple

Let $G$ be a group of order 24, and we shall assume there there exist a non-normal 2-sylow group in $G$. i want to show that it's not simple. first i have showed that there are exactly three 2-sylow ...
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2answers
87 views

There is no simple group of order $144$

There is no simple group of order $144$ I have a question to the proof of the statement above (from the book J. Gallian, Contemporary abstract algebra), it is about the index theorem, so I give ...