## What is the order of a permutation group?

The order of a group (of any type) is the number of elements (cardinality) in the group. By Lagrange’s theorem, the order of any finite permutation group of degree n must divide n! since n-factorial is the order of the symmetric group Sn.

## How many ways can 4 students line up for a photo?

There are 4! ways to line up since there are 4! permutations = 4*3*2*1 = 24 different ways of lining up.

**What are the 3 types of permutation?**

Permutation can be classified in three different categories:

- Permutation of n different objects (when repetition is not allowed)
- Repetition, where repetition is allowed.
- Permutation when the objects are not distinct (Permutation of multi sets)

**Is posing for a picture permutation?**

It is not a combination lock it is a permutation lock. If it truly were a combination lock then if 5–12–32 opens your lock so would 5–32–12, 12–5–32, 12–32–5, 32–5–12, and 32–12–5. In posing for pictures I would say the order matters, so it is a permutation.

### What is A_N group?

For n > 1, the group An is the commutator subgroup of the symmetric group Sn with index 2 and has therefore n!/2 elements. It is the kernel of the signature group homomorphism sgn : Sn → {1, −1} explained under symmetric group. The group An is abelian if and only if n ≤ 3 and simple if and only if n = 3 or n ≥ 5.

### How many ways can 7 students finish a race in 1st 2nd and 3rd place?

210 different ways

The answer is 210. There are 210 different ways these seven runners could take first, second, and third place in the race.

**How many ways can a family of 8 be arranged in a line for a photograph?**

∴ There are 40,320 possible ways 8 students can be arranged in a line.

**How do you find the order of permutation examples?**

Order of Permutation Group

- Order of Permutation-: For a given permutation P if Pn= I (identity permutation) , then n is the order of permutation.
- Example 1-: How many times be multiplied to itself to produce.
- Solution-: Let P=
- Then P2=P.P=
- P2=
- P3= P2.P=
- P3= =I.
- Order=3.

#### What is the order of 123 )( 45?

(f) Each 3-cycle is disjoint from exactly one transposition, so there are 20 permu- tations conjugate to (123)(45). The order of the centralizer of this permutation is 120/20 = 6.

#### How many ways can a photographer at a wedding arrange 5 people in a row including the bride and groom if the the bride and groom must stand next to each other?

Now you have six “objects” to arrange: the five individuals and the bride/groom group. There are 6! = 720 ways to do that.

**What is a permutation vs combination?**

A permutation is an act of arranging the objects or numbers in order. Combinations are the way of selecting the objects or numbers from a group of objects or collection, in such a way that the order of the objects does not matter.

**What is degree of permutation group?**

The degree of a group of permutations of a finite set is the number of elements in the set. The order of a group (of any type) is the number of elements (cardinality) in the group. By Lagrange’s theorem, the order of any finite permutation group of degree n must divide n! since n-factorial is the order of the symmetric group S n.

## How to prove the formula for permutation?

– First object can be chosen in n ways – Second object is chosen in n-1 ways – Third object can be chosen in n-2 ways – Similarly rth object can be selected in ways – So all r objects can be chosen in ways – For above divide and multiply by (n-r)! – this is the one of the simplest proof of permutations

## How does one compute the sign of a permutation?

The sign of a permutation Theorem 11.1. Suppose n 2. (a) Every permutation in Sn is a product of transpositions. (b) If the identity I = ⌧ 1…⌧r in Sn is expressed as product of transpositions, r must be even. Before giving the proof, we need the following lemmas. Lemma 11.2. Suppose a,b,c,d 2{1,…,n} are mutually distinct elements.

**How do you find the permutation?**

Choose “Count permutations” as the analytical goal.