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
3
votes
0 answers

$H$ is a cyclic group of order $p^m$.

I am reading "A Course in $p$-adic Analysis" by Alain M. Robert. I have found the following statement (page 60): Let $\sigma \colon C_{p^m} \rightarrow C_{p^{m-1}}$ be a surjective homomorphism between cyclic groups of order $p^m$ and $p^{m-1}$,…
HeMan
  • 3,119
3
votes
1 answer

Rotman introduction to theory of groups exercise

From Rotman "Introduction to the Theory of Groups", ex. 2:54: Let $ G $ be a finite group, and let $H$ be a normal subgroup with $(H,[G:H])=1$. Prove that $H$ is the unique such subgroup in G. That exercise was introduced here before: link It's…
Matthewr
  • 31
3
votes
3 answers

Order of a Group Based on Their Exponents

Let $G$ be a finite commutative group. The exponent of $G$ is the least common multiple of all orders of all the elements of $G$. Show that $G$ has an element whose order equals the exponent of $G$. Here is the question that I have been stuck on…
user425349
3
votes
1 answer

Is there any finite group G such that G has no subgroup of order $d_1$, $d_2$ but have a subgroup of order $d_1 d_2$

Is there any finite group G with $d_1$, $d_2$ and $d_1\times d_2$ are proper dividors of $|G|$ such that G has no subgroup of order $d_1$, $d_2$ but have a subgroup of order $d_1\times d_2$ $S_5$ has subgroup $A_5$ of order 60 but does not have…
SSSM
  • 117
3
votes
1 answer

right Sylow statement?

I would like to know if this statement (which i just met and suspiciously never realized before) and its proof are true: Let $p$, $q$ be distinct primes and $G$ a group of order $n=p^{\alpha}q^{\beta}$. If $H$ is a $p$-Sylow subgroup of $G$ and $K$…
Louis La Brocante
  • 2,109
3
votes
3 answers

Group of arbitrary order

Can we construct a group of order $n$ for any $n \in \mathbb{Z}^+$ i.e set of positive integers? Are there theorems which characterize the order of any finite group? What is the smallest possible restriction you can have on a group such that there…
DurgaDatta
  • 461
3
votes
1 answer

Is there any finite non trivial Group with this property?

I was asked to have a look at a problem: There is no a finite non-trivial group $G$ that all its non-trivial elements can be commuted with exactly half elements of group . For the first step, I saw I could not prove it directly so, I assumed we…
Mikasa
  • 67,374
2
votes
0 answers

SO(3,q) to PGL(2,q)

Can anyone suggest a reference to an explicit formula giving, in the standard matrix notation, a homomorphism from SO(3,q) to PGL(2,q) (classical matrix groups: orthogonal of dimension 3 and projective general linear of dimension 2 over finite field…
Alexandre Borovik
  • 301
2
votes
0 answers

structure of a p-group

Let $G=\langle x_1,\ldots,x_6\rangle$ be a non abelian group of order $p^6$ and exponent $p$. Also we know that $[x_i,x_j]=1$, for $1\leq i, j\leq6$, except $[x_1,x_5]=x_3$, $[x_1,x_6]=x_4$ and $[x_2,x_6]=x_4$. I want to find structure of such a…
lura
  • 21
2
votes
0 answers

How can we find element orders of a finite group?

Suppose $p$ and $q$ are two prime divisor of the order of a finite group $G$. I want to know if $G$ has an element of order $pq$ using the character table of $G$. Is this possible? If so, please suggest a method.
Sahar rad
  • 49
2
votes
1 answer

Conjugacy classes of two element in a group with cyclic Sylow subgroup

Let $G$ be a finite group such that Sylow $p$-subgroup $G$ has order $p$. Let $x$ and $y$ be two elements of order $p$. Is true that $x$ and $y$ are conjugate in $G$?
User1257
  • 43
2
votes
0 answers

Sort-of-multiplicative functions on the group algebra

Let $G$ be a finite group. Which functions $f:G \to \mathbf{C}$ obey the equation $$ \sum_{g \in C_1,h \in C_2} f(gh) = \left(\sum_{g \in C_1} f(g) \right)\left(\sum_{h \in C_2} f(h)\right) $$ for all conjugacy classes $C_1,C_2 \subset G$? Can you…
Sweet Potatum
  • 41
2
votes
1 answer

Proof of the First Sylow Theorem in Herstein's $Abstract$ $Algebra$

I am reading the following proof of Sylow's First Theorem given in Herstein's Abstract Algebra: Suppose $G$ is a group of order $p^n m$ where $p$ a prime and $p$ does not divide $m$. Then $G$ has a subgroup of order $p^n$. Herstein proceeds by…
user38268
2
votes
1 answer

Determinantal order of character of a group.

The notion of determinantal order can be found in 'Character Theory of finite groups' by I Martin Isaacs. If $\chi$ be a linear character of a finite group G, show that the order of $\chi$ in the group of linear characters of G is…
nabajit
  • 21
2
votes
2 answers

Finite groups and one-to-one functions on them.

I am having trouble with this problem: Assume that $(\mathbb{G}, *)$ is a finite group and there exists a positive integer $n$ such that gcd($n, |\mathbb{G}|)=1$. Prove that the function $F_n: \mathbb{G} \rightarrow \mathbb{G} $ defined $(\forall…
L.J.
  • 43
  • 5
1 2
3
…
12 13