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 \}. |