5. Cosets & Lagrange's Theorem

Question 1

Coset

Answer 1

a set of all the products obtained by multiplying each element of a subgroup in turn by one particular element of the group containing the subgroup

Question 2

Coset Property

Answer 2

1. 2 left cosets of a subgroup H of a group G are either equal or disjoint.
2. G is the disjoint union of all the left cosets of H, the left cosets of H form a partition of G.

Question 3

Lagrange’s Theorem

Answer 3

G is a finite group and H is a subgroup of G. The order of H divides the order of G

Question 4

Index of H in G

Answer 4

The common value of |G/H| and vice versa

Question 5

Finite & Infinite Order of an element

Answer 5

Smallest integer n>0 such that x^n=e. If it doesnt have finite order, it has infinite order.

Question 6

Properties of Order of an Element

Answer 6

1. Every element in a finite group has finite order, and the order of every element divides the order of the group
2. A group of prime order is cyclic
3. In a finite group, the number of elements of prime order p is divisible by p − 1

Question 7

Conjugacy

Answer 7

a and b are conjugate if there exists a g in G such that a=gbg^-1

Question 8

Conjugacy Class

Answer 8

The equivalence class in G under conjugacy