All Questions
29
questions
18
votes
1
answer
6k
views
What is the number of Sylow p subgroups in $S_p$?
I am reading the Wikipedia article entitiled Sylow theorems. This short segment of the article reads:
Part of Wilson's theorem states that
$(p-1)!$ is congruent to $-1$ (mod $p$) for every prime $p$....
9
votes
1
answer
2k
views
What is the largest abelian subgroup in $S_n$?
I am almost to complete my first course in group theory. I have read Dummit and Foote into chapter 5.
I know that I can always find an abelian subgroup isomorphic to $C_{k_1} \times C_{k_2} \times .....
6
votes
1
answer
178
views
Choosing elements from an abelian group $\mathbb{Z}_n$ that make the enumeration of partitions incomplete.
Take an abelian group $(\mathbb{Z}_n,+)$ and enumerate all partitions of two elements (i.e. $x=x_1+x_2$) of each element $\{0,1,...,n-1\}=\mathbb{Z}_n$. Take, for example, abelian groups $\mathbb{Z}_9$...
5
votes
4
answers
618
views
${{p-1}\choose{j}}\equiv(-1)^j \pmod p$ for prime $p$
Can anyone share a link to proof of this?
$${{p-1}\choose{j}}\equiv(-1)^j(\text{mod}\ p)$$ for prime $p$.
5
votes
2
answers
4k
views
Maximum possible order of an element in $S_7 \text{ and } S_{10}$
Exercise :
Find the maximum possible order of an element of the group of permutations $S_7$. Do the same thing for $S_{10}$.
Discussion :
Recalling that any permutation can be written as a ...
5
votes
2
answers
166
views
How to prove $\frac{1}{q} \binom{p^k q}{p^k} \equiv 1 \mod p\;\;$?
An argument that I just came across asserts that if $p$ is prime, and $k, q\in \mathbb{Z}_{>0}$, then
$$\frac{1}{q} \binom{p^k q}{p^k} \equiv 1 \mod p \;.$$
This assertion was made in a way (i.e. ...
4
votes
2
answers
1k
views
How many rings are there for a given order?
Often I have encountered questions like :
How many rings of order 4 are there upto isomorphism?
Often the solution involved brute-force treatments as checking the multiplication tables.
But it is ...
4
votes
0
answers
109
views
How many connected nonisomorphic graphs of N vertices given certain edge constraints?
Background:
I’m helping a colleague with a theoretical problem in ecology, and I haven��t quite the background to solve this myself. However, I can state the problem clearly, I think:
Problem statement:...
3
votes
2
answers
953
views
Permutations minus Transpositions
I want a formula that allows me to find all the permutations in $S_n$ (which is the set of all the integers from 1 to $n$) which don't contain a transposition.
Attempt:
Lets call $g(n)$ the ...
3
votes
1
answer
283
views
Symmetry in the set of integers that cannot be written as $ap+bq$ where $a,b$ are non-negative integers for relatively prime $p,q$
I was studying symmetries (as an introduction to Group Theory) and found this question-
Let $p,q$ be relatively prime positive integers and let $X$ be the set of integers that cannot be written as $...
3
votes
0
answers
86
views
A question on a possible cyclic sieving phenomenon
Let $G$ be a finite group. Consider the set $X_G:=\cup_{H\le G} G/H$, where the disjoint union is taken over all left cosets of the subgroups $H \le G$. Let $S \subset G$ be a generating set for $G$ ...
3
votes
0
answers
82
views
'Randomness' of inverses of $(\mathbb{Z}/p \mathbb{Z})^\times$
Suppose you are given the group $(\mathbb{Z} / p \mathbb{Z})^{\times}$, where $p$ is prime. Let $A_p$ denote the sequence whose $j$th element is the inverse of $[j]$. For instance, if $p = 7$, the ...
2
votes
1
answer
211
views
Select k no.s from 1 to N with replacement to have a set with at least one co-prime pair
Given $1$ to $N$ numbers. You have to make array of $k$ no.s using those no.s, where repetition of same no. is also allowed, such that at least one pair in that chosen array is co-prime. Find no. of ...
2
votes
0
answers
276
views
When the 24 Game is solvable given the four cards?
Consider the 24 Game with a deck of cards that forms a set $ \Omega = \{1, 2, ..., 13\}$, and we try to compute 24 using addition, subtraction, division and multiplication in Rational numbers $\mathbb{...
2
votes
0
answers
99
views
Generalization of Cauchy-Davenport inequality to $\mathbb Z_p^d$
Is there some generalization of Cauchy-Davenport inequality to the group $\mathbb Z_p^d$? ($p$ prime number, $d \ge 3$)
For example, Kneser Theorem says that if $G$ is any abelian group and $\...