Order of a Group, Cosets, definition,Examples

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

Examples:

Order of a Group
  • Dihedral group [katex]D_2n[/katex] has order [katex]2n[/katex]
  • Symmetric group [katex]S_3[/katex] has order [katex]6[/katex]
  • [katex]A[/katex]={[katex]1,2,3,4,5[/katex]} modulo [katex]6[/katex] has order [katex]6[/katex].

Order of an element:

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

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

Cosets of subgroup H in group G:

Let [katex]H[/katex] be a subgroup of a group [katex]G[/katex]. If [katex]a \in G[/katex], the right coset of [katex]H[/katex] generated by [katex]a[/katex] is, [katex]Ha[/katex] = { [katex]ha, h ∈ H[/katex] };
and similarly, the left coset of [katex]H[/katex] generated by [katex]a[/katex] is [katex]aH[/katex] = { [katex]ah, h ∈ H[/katex] }

Example:

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

 The left cosets of [katex]H[/katex] in [katex]G[/katex] are,
[katex]eH = e*H = \{e* h | h ∈ H\} = \{ 0+h| h ∈ H,\}= \{0, 2\}[/katex].
[katex]1H= 1*H = \{ 1^* h | h ∈ H \} = \{ 1+h| h ∈ H\} = \{1, 3\}[/katex].
[latex]2H=2\ast H=\{2\ast h\vert h\in H\}=\{2+h\vert h\in H\}=\{0,2\}[/latex].
[katex]3H= 3*H = \{3 * h |h ∈ H \} = \{ 3+h| h ∈ H \} = \{1, 3 \}[/katex].
Hence there are two cosets, namely [katex]0*H= 2*H = \{0, 2 \} and 1*H= 3*H = \{1, 3 \}[/katex].

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