All Questions
29
questions
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 ...
- 21.4k
0
votes
1
answer
430
views
Find the count of unique subset sums in a powerset
Given a list of integers, I want to find the count of unique sums for all possible subsets of length $N$.
I don't want to know the sum, or the subsets, just the count of possible unique sums for ...
- 149
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{...
- 268
0
votes
1
answer
81
views
Condition that for a given set of numbers and given divisor all finite sums from this set contain all possible remainders
Given $q \in \mathbb{N}$ and ${a_1, a_2, ...}$ where each $a_j \in \mathbb{N} \cup{\{0\}}$ define $A_p=$ {set of all finite sums of $\{a_1 ... a_p\}$ such that each $a_j$ will appear either $1$ or $0$ ...
- 633
0
votes
3
answers
63
views
Proof about elementary group theory
Let $X = \{1, 2, 3, 4, 5\}$ and $σ : X → X$ be a bijective function. 1) Prove
that $σ^n$ is invertible for all $n ∈ N$. 2) Prove that there is an natural number
$n$ such that $σ^n = I_x$. Hint: Use ...
- 2,031
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 .....
- 3,711
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 $p$....
- 3,711
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 ...
- 857
0
votes
0
answers
134
views
Zero-Sum Partitions of Nonzero Elements of a Ring
In this question, rings are not necessarily finite nor do they need to be unital (i.e., the multiplicative identity may not exist). Although I shall almost exclusively discuss finite commutative ...
- 49.8k
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 ...
- 47
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 $\...
- 1,006
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 ...
- 269
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. ...
- 14.5k
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 $p$.
- 4,793