Subgroup and Order of Group | Mathematics

In mathematics, a group is a fundamental algebraic structure consisting of a set of elements combined with a binary operation that satisfies four key properties: closure, associativity, identity, and invertibility.

An example of a group is the set of integers under addition. In this case, the binary operation is addition (+), the identity element is 0 (since adding 0 to any integer returns the same integer), and the inverse of any integer a is its negative −a (since a + (−a) = 0).

In this article, we will discuss subgroups and the order of a group, which are fundamental parts of group theory.

What are Subgroups?

A nonempty subset H of the group G is a subgroup of G if H is a group under the 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

Read More about Groups.

Examples of Subgroup

Some examples of subgroups are listed as follows:

• Integers under Addition (Z, +): The set of even integers is a subgroup of the group of all integers under addition.
• Modular Arithmetic: In modular arithmetic, the set of integers modulo n forms a group, and the set of integers that are multiples of a divisor d of n forms a subgroup.

Properties of Subgroups

Some of the common properties of subgroups are:

• G is a subgroup of itself and {e} is also a subgroup of G, these are called trivial subgroups.
• Subgroup will have all the properties of a group.
• A subgroup H of the group G is a normal subgroup if g^-1Hg = 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 may or may not be a subgroup.

Types of Subgroups

Some of the common types of subgroups are:

• Trivial Subgroup
• Proper Subgroup
• Cyclic Subgroup
• Normal Subgroup
• Cosets

Trivial Subgroup

Every group has two trivial subgroups, the subgroup containing just the identity element and the group itself. These are the smallest and largest subgroups, respectively.

Proper Subgroup

A subgroup that is not equal to the group itself is called a proper subgroup. It contains at least one element other than the identity element.

Cyclic Subgroup

Generated by a single element of the group. If a is an element of a group G, then the cyclic subgroup generated by a, denoted ⟨a⟩, is the set {aⁿ : n∈Z}.

Normal Subgroup

A subgroup H of a group G is normal if gHg⁻¹ = H for all g ∈ G. Normal subgroups play a crucial role in the theory of quotient groups

Cosets

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

Order of a group is the number of elements in the group. Similarly, the order of a subgroup is the number of elements in that subgroup, which is always less than or equal to the order of the original group.

Order of a group (G) is the number of elements present in that group, i.e it's cardinality. It is denoted by |G|.

For finite groups, the order is simply the count of elements in the group. For example, if a group has 5 elements, its order is 5.

Order of Element

In group theory, the order of an element in a group is the smallest positive integer n such that raising the element to the power of n results in the identity element of the group.

For an element aaa in a group G, the order of a, denoted as ∣a∣ or ord(a), is defined as:

∣a∣ = min{n∈Z⁺ : aⁿ=e}

Where e is the identity element of the group G.

Note: If there is no such positive integer n, then the order of the element is considered infinite.

Properties of Order of Element

Common property of order of element are:

• 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.

Lagrange’s Theorem

Statement of Lagrange’s theorem is:

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

Related GATE Questions:

• Gate CS 2018
• Gate CS 2014 (Set-3)

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