**Order of a Group:** The order of group G is the number of elements present in that **group** G, also say it’s cardinality. It is denoted by \left|G\right| or O(G).

**Examples:**

**Dihedral group D_2n**has order 2n**Symmetric group S_3**has order 6- A={1,2,3,4,5} modulo 6 has order 6.

**Order of an element:**

**Order of element** a\in G is the smallest positive integer n such that a^n=e, where e denotes the identity element of the **group**, and a^n denotes the product of n copies of a.The order of every element of a finite group is finite.

- The Order of an element a of group is the same as that of its inverse a^{-1}.
- If a is an element of order n and p is prime to n, then a^p is also of order n.
- Order of any integral power of an element b cannot exceed the order of b.
- If the element a of a group G is of order n, then a^k=e
**if and only if**n is a divisor of k. - The order of the elements a and x^{-1}ax is the same where a, x are any two elements of a group.
- If a and b are elements of a group then the order of
**ab**is same as order of**ba**.

**Cosets of subgroup H in group G:**

Let H be a subgroup of a group G. If a \in G, the right coset of H generated by a is, Ha = { ha, h ∈ H };

and similarly, the left coset of H generated by a is aH = { ah, h ∈ H }

`Example:`

Consider Z_4 under addition (Z_4, +), and let H={0, 2}. e = 0, e is identity element. Find the left cosets of H[katex] in [katex]G?**Solution:**

The left cosets of H in G are,

eH = e*H = \{e* h | h ∈ H\} = \{ 0+h| h ∈ H,\}= \{0, 2\}. |

1H= 1*H = \{ 1^* h | h ∈ H \} = \{ 1+h| h ∈ H\} = \{1, 3\}. |

[latex]2H=2\ast H=\{2\ast h\vert h\in H\}=\{2+h\vert h\in H\}=\{0,2\}[/latex]. |

3H= 3*H = \{3 * h |h ∈ H \} = \{ 3+h| h ∈ H \} = \{1, 3 \}. |

Hence there are two cosets, namely 0*H= 2*H = \{0, 2 \} and 1*H= 3*H = \{1, 3 \}. |