SubGroup -Types and Examples

subgroup is defined as Let (G,*) be a group and H be a non-empty subset of G.if H is itself a group with the same binary operation *. Then H is called a subgroup of G.

If H is a subgroup of a group G . Then is denoted by H≤G

For example:

1. (z,+) is a sub-group of (Q,+)
2. (Q,+) is a subgroup of (R,+)

What is Subgroup?

Let G={1,-1, I, -i} be grouped under the binary operation of multiplication.

Let H={1,-1} be a non-empty subset of group G.

Here H satisfies all the conditions of a group same binary operation (multiplication) as defined in G.

1. Closure
2. Associative
3. Existence of the identity element
4. Existence of inverse

So, H is a subgroup of G.

Types of Subgroup

There are two types of subgroup

1. Trivial subgroup/improper subgroup
2. Non–trivial subgroup/proper subgroup

Trivial subgroup

Every group G has at least two subgroups.

1. G itself
2. Identity subgroup {e}

These are called trivial subgroups.

For example:

Let G ={1,-1,i,-i} than (G,.) is group.

A trivial subgroup of G. are

1. G
2. {1}

Non–trivial subgroup

Any subgroup of G, other than two trivial subgroups is called non-trivial subgroups of G.

For example

Let G ={1,-1,i,-i} be a group under the binary operation of multiplication.

Let

H={1} and k={1,-1}

M={1,-1,i,-i}

H and M are the trivial subgroup of G.

K is a non-trivial subgroup of G.

For example:

G={e,a,b,c} defined by

.	e	a	b	c
e	e	a	b	c
a	a	e	c	b
b	b	c	e	a
c	c	b	a	e

Solution:

It is clear from table

a²=b²=c²=e

Here subgroup of G

1.  {e}
2. {e,a}
3. {e,b}
4. {e,c}
5. G

{e} ,G are trivial subgroup of G

{e,a},{e,b},{e,c} are non-trivial subgroup of G.

Important point of subgroup

1. H is a subset of G
2. H is a group
3. H and G use the same binary operation

Frequently Asked Question-FAQs