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.

learn more… | top users | synonyms

3
votes
1answer
24 views

Find\construct a group of order $q(q-1)$ s.t. …

Problem- Let $q$ be a power of a prime $p$ say $q=p^k$. Show that there exists a group $G$ of order $q(q-1)$ with a normal elementary abelian subgroup of order $q$ and such that all elements of order ...
3
votes
2answers
35 views

Show that every proper subgroup of this group is finite.

Let $G$ be the group of rational numbers in $[0,1)$ whose denominator is a power of $2$: \begin{align*} G &= \{r/2^k : \text{$r \in \mathbb Z$, $0 \le r < 2^k$, $k = 0, 1, ...
-4
votes
0answers
37 views

To prove any two non-abelian groups of order $pq$ are isomorphic , without Sylow theorems [on hold]

How to prove , without using Sylow theorems , that any two non-abelian groups of order $pq$ , where $p,q$ are distinct primes , are isomorphic ?
0
votes
0answers
32 views

last part of proof of schur zassenhaus theorem.

Theorem states- Let $G$ be a finite group of order $mn$ and $N$ be a normal subgroup of order $n$, then schur zassenhaus states that there exist a complement of $N$ in $G$ of order $m$ and all such ...
0
votes
0answers
27 views

Centralizer of unique cyclic subgroup of order equal to exponent of group [on hold]

Let $G$ a finite group and $K\leq G$ the unique cyclic subgroup of $G$ with $|N|=\exp(G)$. Is $C_{G}(N)=N$?
1
vote
1answer
32 views

Consequence of Lemma: If G is abelian with exponent n, then $|G|\big\vert n^m$ for some $m\in N$

Lemma: If G is abelian with exponent n, then $|G|\big\vert n^m$ for some $m\in N$. Theorem to be proved: Suppose G is finite abelian and group of order m, let p be a prime number dividing m. Then G ...
0
votes
1answer
20 views

How to find orbits and isoropy group?

About this problem ${a}$, I am wondering if there are 5 orbits in $A$? The 5 orbits separately contain elements which 3 are all the same, 2 of 3 are the same and all 3 are different? I am confused ...
2
votes
2answers
69 views

Prove that $n_p(N)$ divides $n_p(G)$

Let $N$ be a normal subgroup of $G$ where $G$ is finite group, then we have to prove $n_p(N)$ divides $n_p(G)$ ( here $n_p(G)$ means number of sylow $p$-subgroups of $G$) I was able to prove that ...
2
votes
1answer
22 views

Group of order $|G|=pqr$, $p,q,r$ primes has a normal subgroup of order

Let $p,q,r$ be positive primes, $p<q<r$, and let $G$ be a group with $|G|=pqr$. Show that there exists a normal subgroup $H$ of $G$ of order $qr$. I've seen this post Groups of order $pqr$ and ...
1
vote
0answers
31 views

Give a $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$

Consider $SL_{2}(\Bbb Z_p)$ if q & p be two primes, $p>q$. Give an example of a subgroup $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$ when i) $q|(p-1)$ ii) $q|(p+1)$
0
votes
0answers
19 views

Conjugacy classes in non-solvable group

Suppose $A$ is an arbitrary subset of group $G$ and $K_G(A)$ be the number of $G$-conjugacy classes contained in $A$. Also suppose $G$ non-solvable group, $N\unlhd G$, $G/N$ is abelian, $|G/N|=6$ and ...
2
votes
1answer
24 views

Elementary abelian $p$-group

How can we show that if $N$ is abelian and $C_G(N)=N$, then $N$ is an elementary abelian $p$-group for some prime $p$?
1
vote
1answer
30 views

Classification of groups of order $p^2q$

I have done the classification of groups of order $p^2q$, where $p>q$. $p,q$ odd distinct prime If $P\in Syl_{p}(G)$ & $Q\in Syl_{q}(G)$ then Case $1$: $P=\Bbb Z_{p^2}$. I have done. But my ...
0
votes
0answers
30 views

Conjugacy classes in non-abelian simple group

Can we say that every non-abelian simple group has at least 4 non- identity classes?
1
vote
1answer
31 views

Question about the answer to Kac's problem: 'Can one hear the shape of a drum?'.

I'm looking at the article of Gordon, Webb and Wolpert http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, having only basic notions of group theory. In this article the authors describe ...
1
vote
2answers
36 views

Subgroups and justification

Which of these subsets are subgroups of the given group and justify your answer. The group $R^+$ of postive reals under multiplication. The subset $H=(3n|$ $n\in Z^+)$. The group of nonzero ...
1
vote
2answers
86 views

Are all torsion groups finite groups?

Are all torsion groups finite groups? I've been trying to find a counter example, but have had no luck so far. Can anyone throw me one, or give me an idea to prove this?
0
votes
1answer
35 views

Groups of Order 2 with subgroups

Let G be an abelian group and $a,b\in G$ be two distinct elements with a and b or order $2$. Show that $H=\{e,a,b,ab\}$ forms a subgroup and write out its multiplication table. Justify why all the ...
1
vote
1answer
22 views

finite semigroup on one generator,cycle, tail,group,zero element

Suppose we have a finite semigroup on one generator. It has a tail of length r and cycle of length c.The cycle is a group, but what can be chosen as a neutral element of it?Why is not ANY element ...
1
vote
1answer
46 views

Solvability of a group

What is the intuition behind the solvable groups? It is defined by composition series. Is there any intuitive way to understand it?
2
votes
1answer
42 views

On a finite group with unique minimal subgroup

EDIT: Let $G$ be a finite group with a unique minimal subgroup. Then we know that $G$ is a $p$-group. Let $p\neq2.$ Is it true that every subgroup $H\le G$ and every quotient $G/N$ have a unique ...
1
vote
1answer
33 views

Sylow subgroups of a non abelian group $G$ with $|G|=21$ and $|G|=39$

I am trying to solve the following exercise: ¿How many Sylow subgroups has a non abelian group $G$ of order $21$ and $39$ respectively. I could do the following: a) $|G|=21=3\cdot 7$. I'll call ...
0
votes
1answer
52 views

Non-isomorphic groups

How to prove that $Z/2\times Z/2$ and $ Z/4$ are not isomorphic? I think that $Z/2\times Z/2$ is not cyclic. Hence $Z/2\times Z/2$ and $ Z/4$ are not isomorphic. Thank you.
3
votes
0answers
27 views

Restricting a representation to a subgroup

This little factoid from algebra quals stumped me: Let $G$ be a finite group and $H \triangleleft G$ an index $2$ subgroup. If we take an irreducible complex representation $V$ of $G$ and restrict it ...
3
votes
0answers
31 views

Product of the elements cannot be 1

I have to show in a group $G$ of order $4k+2$ the product of the elements cannot be $1$. What I got so far is that there exists a subgroup $H$ of index $2$ in $G$ (considering left regular ...
2
votes
2answers
41 views

Group theory and group representation

I am fairly new to group theory and representation. I am currently looking at faithful representations. I am not quite sure what is the "use" of a faithful representation. I cannot find any "easy to ...
3
votes
0answers
36 views

Abelian group action exercise

Let $X$ be a set with $n$ elements and let $G$ be an abelian group acting on $X$ such that: $$(i) \space gx=x \space \forall x \implies g=1,$$ $$(ii) \space \forall x,y \in X, \exists g: gx=y.$$ Show ...
3
votes
1answer
43 views

There are no simple groups of order $27p$

Statement Show that there are no simple groups of order $27p$ for any $p$ prime number. I got stuck with this problem, I'll write what I've done so far: Suppose $G$ is a group with $|G|=3^3p$, $p$ ...
0
votes
2answers
38 views

Give an example of three different groups with eight elements

Give an example of three different groups with eight elements. Why are the groups different? One particular answer that I found was the groups $\mathbb{Z}_8$, $\mathbb{Z}_4 \times\mathbb{Z}_2$, and ...
11
votes
2answers
318 views

Is a group uniquely determined by the sets {ab,ba} for each pair of elements a and b?

For a given group $G=(S,\cdot)$ with underlying set $S$, consider the function $$ F_G:S\times S\to\mathcal P(S)\\ F_G(a,b):=\{a\cdot b,~b\cdot a\} $$ from $S\times S$ ...
1
vote
1answer
45 views

Sylow subgroups problem

Let $G$ be a finite group and $p<q$ such that $p^2$ doesn't divide $|G|$. Let $H_p$ and $H_q$ be Sylow subgroups of $G$ with $H_p \lhd G$. Show $H_pH_q \lhd G \space \implies H_q \lhd G$. From the ...
1
vote
1answer
37 views

Sylow p-subgroups and Sylow theorem

Find all Sylow 3-subgroups of $S_3\times S_3$? This is what I already found: Since $O(S_3\times S_3)=36=2^2 3^2$ Sylow- $3$ subgroups have order $9$. If $n_3$ is the no. of Sylow- $3$ subgroups, Then ...
0
votes
1answer
24 views

Is this set a generating set for this (normal) subgroup?

Suppose $G$ is a finite group and $N$ is a normal subgroup of $G$. Suppose also that $X$ is a generating set for $G$, and that $Y$ generates $N$ as a normal subgroup of $G$ (i.e. $N$=$\langle ...
3
votes
2answers
86 views

Presentations representing different groups.

Using GAP, I knew that groups $G=\langle x,y;x^4,x^2y^2,xyxy^{-1}\rangle$ and $H=\langle x,y;x^4,y^4,xyxy^{-1}\rangle$ are different. But I want to prove it. I tried to do something using Tietze ...
1
vote
0answers
47 views

Groups of orders 108, 120, 144, 168, 180, 228 and 240 [closed]

Prove that every group of order $\in \{ 108, 120, 144, 168, 180, 228\}$ has a subgroup of order or index $6$. But, this is not true for all groups of order $240$.
0
votes
1answer
26 views

Why if H⊴G and H is solvable and G/H is solvable, then G is also solvable?

Plus, to the proof any p-group G is solvable,although Z(G) is normal to G and which is abelian, but G/Z(G) is not abelian, so what is the chain subgroup to show that G is solvable?
1
vote
2answers
45 views

$N$ a normal subgroup of $G$ with $|G|$ odd and $|N|=5$ satisfies $N \subset Z(G)$

Exercise Let $N$ be a normal subgroup of $G$. Suppose that $|N|=5$ and that $|G|$ is odd. Show that $N \subset Z(G)$. I am sorry for not writing any work of mine but I really don't know where to ...
1
vote
1answer
59 views

I want to form a group of order one, is it possible?

Can a set with one element be a group, or does a group need to have at least two elements?
2
votes
1answer
26 views

Calculating number of conjugacy classes for a prime power group $G$

$\space$Let $p$ be a prime number and let $G$ be a group of order $p^4$ such that $|Z(G)|=p^2$. Calculate the number of conjugacy classes of $G$. Since $Z(G) \neq G$, then $G$ is not abelian. I'll ...
1
vote
1answer
41 views

$x \rightarrow x^n$ is a group automorphism of a finite abelian group G [closed]

How do we prove that the map $\phi:G \rightarrow G$ defined by $\phi(x) = x^n$ for some $n \geq 0$ is a group automorphism of $G$ if $\gcd(|G|,n)=1$?
1
vote
0answers
39 views

How do I get the generators of a group formed by combining two groups with known generators?

Consider two groups, $G$ and $H$, with generating sets $S$ and $T$, respectively. (That is, $G=\langle S \rangle$ and $H=\langle T \rangle$.) Let us say that we can represent elements of both $G$ ...
2
votes
2answers
40 views

If $|G|<\infty$ and $H\leq G$ is such that $[G: H]=2$ then $|x^G|=|x^H|$ or $|x^H|=\frac{1}{2}|x^G|$ for all $x\in H$?

Let $G$ be a finite group and $H$ a subgroup of $G$ with index $2$, that is, $[G: H]=2$. Recall that $$C_H(x)=H\cap C_G(x), $$ where $C_G(x)=\{g\in G: gx=xg\}$. How can I use the second isomorphism ...
1
vote
0answers
28 views

$G$ a finite group $n$-abelian goup and g.c.d.$\big(|G|,n(n-1)\big)=1$ , then to show $G$ is abelian [duplicate]

Let $G$ be a finite group and $n$ be a given positive integer such that $(ab)^n=a^nb^n , \forall a,b \in G$ and g.c.d.$\big(|G|,n(n-1)\big)=1$ , then how to prove that $G$ is abelian ? If I can show ...
1
vote
1answer
61 views

Complement but not direct summand; Lam, Lectures on Modules and Rings, Example 6.17(5)

Let $S$ be a submodule of an $R$-module $M$. A submodule $C⊆M$ is said to be a complement to $S$ (in $M$) if $C$ is maximal with respect to the property that $C∩S=0$. (This does exist by Zorn Lemma.) ...
3
votes
1answer
48 views

Transitive action on two sets! [duplicate]

Suppose $G$ is a finite group and G acts transitively on sets $X$ and $Y$. Let $a$ and $b$ belongs to $X$ and $Y$ respectively and $G_{a}$ be stabilizer of $a$ in $X$ and $G_{b}$ be stabilizers of ...
1
vote
0answers
39 views

Intuition fail with normal subgroups

My intuition fails me, when I try to undestand normal subgroups. I read that Alternating group $A_n$ is simple for all $n \geq 5$, $n$ is the order of the group. So $A_4$ is possibly (this has to do ...
1
vote
2answers
43 views

Group of order $pq$ with $p\not\mid (q-1)$

Let $p, q$ be prime numbers, with $p<q$. If $G$ is a group of order $pq$ and $p\not\mid (q-1)$, then $G\cong \mathbb{Z}/pq\mathbb{Z}$. The standard way to prove this fact is using Sylow theorems, ...
0
votes
0answers
23 views

Crossed homomorphism: $\varphi(x)=\varphi(y)\iff Kx=Ky$

Let $G$ be a finite group, $N\unlhd G$. A crossed homomorphism from $G$ to $N$ is defined as a $\varphi:G\to N$ s.t. $\varphi(xy)=\varphi(x)^y\varphi(y)$. It's not in general a group homomorphism. ...
3
votes
2answers
83 views

Field that contains $(\mathbb{Z}/p^n\mathbb{Z})^*$

Let $p$ be an odd prime. There exists a field $F$ that contains an isomorphic copy of $(\mathbb{Z}/p^n\mathbb{Z})^*$ as a multiplicative subgroup? Clearly if $n=1$ we can take $F=\mathbb{F}_p$, but ...
3
votes
1answer
54 views

An equality from Representation Theory

Studying Representation Theory of finite groups I've bumped in the following identity: $$\frac{n(n+1)}{2}=\sum_{i=1}^n\frac{(2i-1)!!(2n-2i+1)!!}{(2i-2)!!(2n-2i)!!}$$ My book suggests to prove it ...