Sub Group with Examples

In this class, We discuss Sub Group with Examples.

The reader should have prior knowledge of the group. Click Here.

Sub Group:

A non-empty subset H of a group G is called a subgroup.

1) H is a group on the same operation defined on G.

2) H has the same identity element

Examples:

[Z +] is a subgroup of [R +]

R is a set of real numbers

Z is a set of integer values

[Z +] satisfies all the group conditions.

closure

associative

the identity element is 0

inverse element exists.

Example 2:

The set of even numbers is a subgroup of [Z +]

Example 3:

The set of non-negative integers is not a sub-group of [Z +]

Zero is not present in positive numbers, so the identity element is missing for the subgroup

Note: For any Group [G *], we have [e *] as a trivial sub-group.