Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [finite-groups]

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.

The questions that you can ask under this tag are of the same type that the questions related to the tag : homomorphisms of finite groups, representation theory, structure of finite groups...

You may refer to this Wikipedia entry for an introduction to this topic.

11774 questions
1
vote
1 answer

Idempotents-Irreducible Characters

Given an finite group $G$ and its group algebra $\mathbb{C}G$ and the order relation on the idempotents of $\mathbb{C}G$ such that $e\leq f$ iff $ef=fe=e$. Is the following true: The irreducible characters are in bijection with the minimal…
João Dias
  • 361
1
vote
2 answers

How to define the multiplicative group generated by a set?

I have the following problem: Let $G=\langle a_1,a_2,\ldots,a_n\rangle$ be the multiplicative group generated by $a_1,a_2,...,a_n$. Prove that if $a_ia_j=a_j a_i$ $\forall i,j\in\{1,2,\ldots,n\}$, then $G$ is an abelian group. I don't understand…
MarianaM
  • 13
1
vote
0 answers

How we can conclude that $p\nmid \sum_{x\in H}|x^G|$ in a group with some elements of order $2p$?

Let $G$ be a finite group such that has some elements of order $2p$, where $p$ is an odd prime. Let $H$ be the set of all elements of order $2p$ in $G$. We can show $G$ acts on $H$ by conjugation. So $H$ cut to partition by orbits of this action.…
user204248
  • 21
1
vote
0 answers

show that $O^{p}(G)$ will generate with all $q$-sylow subgroups of $G$ .

suppose that $G$ is finite group and $p$ is a prime number,then show that $O^{p}(G)$ will generate with all $q$-sylow subgroups of $G$ where $q$ is arbitrary prime number and $q\neq p$ . ($O^{p}(G)$ is intersection of all normal subgroups of $G$…
kpax
  • 2,911
1
vote
0 answers

Find generators in GF(19)

I have 2 questions. Finding generators in GF(19) is similar to finding generators in GF(2^p)? Is primitive polynomial needed to find generators for GF(19)? Thanks a lot. Ya Ali.
Mohammad Reza Ramezani
  • 141
0
votes
1 answer

Group of order 105 as a product

I can't solve this problem, please help me! Every group of order 105 is isomorphic to $\mathbb{Z}_5\times H$ where $H$ is a group of order 21.
Creyesm
  • 1
0
votes
2 answers

Is there a 20-order abelian subgroup of $S_5$?

The title says it all: Is there an abelian subgroup of order 20 of $S_5$, the group of permutations of five elements? Thanks for reading!
Larara
  • 1,492
0
votes
2 answers

Group Theory (Abstract Algebra) question! o(G) vs o(g)??

If G is a group, and g is an element of G, what is the difference between the following two notations o(G) and o(g)?
wirdis
  • 3
0
votes
0 answers

Composition Factors of $C_p\times C_p$

I have question that asks me to find the composition series of $C_p\times C_p$, now these are all isomomrphic to the series $\{1\}\lhd C_p \lhd C_p\times C_p$ but the questions wants all the series explicitly and so my solution was as follows: Let…
john smith
  • 395
0
votes
1 answer

Are all the groups of order $n$ contained in $S_n$?

I want to know if I can considere any group of order $n$ is isomorphic to one of $S_n$. Is that true? I can't find a proof.
MaríaCC
  • 474
0
votes
1 answer

general linear groups and definitions

We have two groups, one of them is automorphisms group of a vector space over GF(2) and another one is the direct product of two automorphism group (they are also over GF(2)). Also, via some computations through GAP, we have the generators of…
Nil
  • 1,306
0
votes
1 answer

Diffie helman on additive group

Given the addivite group G, with |G| = n and generator = g, what are the computations and exchanged messages for Diffie-Hellman? I'm not sure because the group is additive.
ralphie91
  • 49
0
votes
2 answers

Suppose G has order 35 and |X|=17

Suppose G has order 35 and |X|=17. Suppose that no point a in X is fixed by all g in G. Find the number of orbits and the size of each orbit. I am not really sure how to go about this, the example I have been given in lectures is not of this form at…
ZZS14
  • 819
0
votes
1 answer

Show that |G|=168 has 8 sylow 7 subgroups and find the normalizer of one of these.

I have shown that there are 8 sylow subgroups, but i dont really know where to start with finding the normalizer. p=7 => n_7={factors of 24}={1,2,3,4,6,8,12,24}congruent 1modp so n_7=1 or 8, but it cannot be 1, hence n_7=8 as required.
ZZS14
  • 819
0
votes
1 answer

if $G$ is a EDP of two finite groups A and B. then order of element $(a,b)\in G=A\times B$ is lcm of order of a and order of b.

If $G$ is a external direct product of two finite groups $A$ and $B$, then order of element $(a,b)\in G=A\times B$ is lcm of order of $a$ and order of $b$.
abc
  • 49
1 2 3
…
12 13