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# SubGroup -Types and Examples

July 18, 2022
written by Azhar Ejaz

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 Hâ‰¤G

For example:

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

Table of Contents

## 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. Nonâ€“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}

### Nonâ€“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

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

### What is the definition of a subgroup?

A subgroup is a smaller subset of a group that has the same qualities as the larger group. So, if Group H is a part of Group G and has all the qualities of a group itself, then Group H is called a subgroup of Group G.

### What makes a subset a subgroup?

A subset of a group that still abides by the rules, or ‘axioms,’ of the original group is referred to as a subgroup. The binary operation must still be consistent with the associativity, closure, inverse, and identity properties for it to be considered a subgroup.

### Types of Subgroup

There are two types of subgroup
Trivial subgroup/improper subgroup
Nonâ€“trivial subgroup/proper subgroup

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