# 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:**

- (z,+) is a sub-group of (Q,+)
- (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.

- Closure
- Associative
- Existence of the identity element
- Existence of inverse

So, H is a subgroup of G.

## Types of Subgroup

There are two types of subgroup

- Trivial subgroup/improper subgroup
- Non–trivial subgroup/proper subgroup

### Trivial subgroup

Every group G has at least two subgroups.

- G itself
- 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

- G
- {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

a^{2}=b^{2}=c^{2}=e

Here subgroup of G

- {e}
- {e,a}
- {e,b}
- {e,c}
- G

{e} ,G are trivial subgroup of G

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

**Important point of subgroup**

- H is a subset of G
- H is a group
- 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

Non–trivial subgroup/proper subgroup

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