Order of a Group, Cosets, definition,Examples

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 $H[latex] in [latex]G$ ?
Solution:

 The left cosets of  in  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\}

Order of an element:

Cosets of subgroup H in group G:

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