# Subgroups

## Subgroups

- A
**subgroup**is a subset of a group that is itself a group. - Not every subset of a group is a subgroup. To be a subgroup, the subset must satisfy all properties of a group:
*closure*,*associativity*,*identity*and*inverse*.

## Subgroup Test

- A simple way to check whether a subset H of a group G is a subgroup is to use the Subgroup Test.
- The
**Subgroup Test**states that a non-empty subset H of a group G is a subgroup if and only if for any two elements a and b in H, the product a(b^(-1)) is in H.

## Examples of Subgroups

- The set of
**even integers**, under the operation of addition, is a subgroup of the group of integers, as it follows all group rules. - The group of
**non-zero rational numbers**under multiplication operation has a subgroup of all rational numbers whose absolute value is 1.

## Properties of Subgroups

- Every group is a
**subgroup of itself**. - The
**criteria for a subset**to be a subgroup are rigorous to ensure that subgroup structure mirrors that of the group. - Subset containing the
**identity element**of the group along with the inverses of all its elements is a subgroup. This is an immediate consequence of the Subgroup Test.

## Cosets and Lagrange’s Theorem

- If H is a subgroup of G, then a
**left coset**is a set formed by multiplying each element of H by a fixed element g of G. - Similarly, a
**right coset**is a set formed by multiplying each element of H by g on the right side, i.e., Hg. **Lagrange’s theorem**is a major result regarding subgroups, which states: in a finite group G, the order (number of elements) of every subgroup H of G divides the order of G. It implies that the number of left cosets (or right sets) equals the order of G divided by the order of H.- Important consequence of
**Lagrange’s theorem**is that the order of a group’s element always divides the order of the group.

Remember, understanding this concept fully is important as it’s a key part of understanding groups, a core aspect of further mathematics.