Subgroup and Order of group | Mathematics

Prerequisite: Groups

Subgroup –

A nonempty subset H of the group G is a subgroup of G if H is a group under binary operation (*) of G. We use the notation H ≤ G to indicate that H is a subgroup of G. Also, if H is a proper subgroup then it is denoted by H < G .

For a subset H of group G, H is a subgroup of G if,
• H ≠ φ
• if a, k ∈ H then ak ∈ H
• if a ∈ H then a^-1 ∈ H

Ex. – Integers (Z) is a subgroup of rationals (Q) under addition, (Z, +) < (Q, +)

Note:
1. G is a subgroup of itself and {e} is also subgroup of G, these are called trivial subgroup.
2. Subgroup will have all the properties of a group.
3. A subgroup H of the group G is a normal subgroup if g ^-1 H g = H for all g ∈ G.
4. If H < K and K < G, then H < G (subgroup transitivity).
5. if H and K are subgroups of a group G then H ∩ K is also a subgroup.
6. if H and K are subgroups of a group G then H ∪ K is may or maynot be a subgroup.

Coset –

Let H be a subgroup of a group G. If g ∈ G, the right coset of H generated by g is, Hg = { hg, h ∈ H };
and similarly, the left coset of H generated by g is gH = { gh, h ∈ H }

Example: Consider Z₄ under addition (Z₄, +), and let H={0, 2}. e = 0, e is identity element. Find the left cosets of H in G?
Solution:

 The left cosets of H in G are,
 eH = e*H = { e * h | h ∈ H} = { 0+h| h ∈ H} = {0, 2}.
 1H= 1*H = {1 * h | h ∈ H} = { 1+h| h ∈ H} = {1, 3}.
 2H= 2*H = {2 * h | h ∈ H} = { 2+h| h ∈ H} = {0, 2}.
 3H= 3*H = {3 * h  |h ∈ H} = { 3+h| h ∈ H} = {1, 3}.
 Hence there are two cosets, namely 0*H= 2*H = {0, 2} and  1*H= 3*H = {1, 3}. 

Order of Group –

The Order of a group (G) is the number of elements present in that group, i.e it’s cardinality. It is denoted by |G|.
Order of element a ∈ G is the smallest positive integer n, such that aⁿ= e, where e denotes the identity element of the group, and aⁿ denotes the product of n copies of a. If no such n exists, a is said to have infinite order. All elements of finite groups have finite order.

Lagrange’s Theorem:

If H is a subgroup of finite group G then the order of subgroup H divides the order of group G.

Properties of the order of an element of the group:
• The order of every element of a finite group is finite.
• The Order of an element of a group is the same as that of its inverse a^-1.
• If a is an element of order n and p is prime to n, then a^p is also of order n.
• Order of any integral power of an element b cannot exceed the order of b.
• If the element a of a group G is order n, then a^k=e if and only if n is a divisor of k.
• The order of the elements a and x^-1ax is the same where a, x are any two elements of a group.
• If a and b are elements of a group then the order of ab is same as order of ba.

Related GATE Questions:
1) Gate CS 2018
2) Gate CS 2014 (Set-3)