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, +)

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

**Note:**

#### 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 Z_{4} under addition (Z_{4}, +), 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 a^{n}= e, where e denotes the identity element of the group, and a^{n} 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.

- 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
^{-1}ax 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**.

**Properties of the order of an element of the group:**

Related GATE Questions:

1) Gate CS 2018

2) Gate CS 2014 (Set-3)

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