Please add test-series name in title

1 vote

4 votes

Best answer

First, you need to check if given group is abelian or not. Here associativity, the existence of an identity, existence of inverse for each element and commutativity are all satisfied so the group is abelian. Now, the order of an element is a smallest positive integer n such that a^n=e. for element a, a^2= a*a=b, a^3= a^2 * a= b*a=c, a^4= a^3*a=c*a=e.

Hence the order of a=4. Option C is correct!

Hence the order of a=4. Option C is correct!

1

@JEET

If $'a'$ is the order of the group, then for smallest $'n'$ we get $a^{n}=e$ where $'e'$ is the identity element.

$(a)^{1}=a$

$(a)^{2}=a*a=b$

$(a)^{3}=a*a*a$

$'*'$ is associative.

So, we can write like this

$(a)^{3}=(a*a)*a=b*a=c$

$(or)$

$(a)^{3}=a*(a*a)=a*b=c$

$(a)^{4}=a*a*a*a=(a*a)*(a*a)=b*b=e$

$(or)$

$(a)^{4}=a*a*a*a=a*(a*a*a)=a*c=e$

$(or)$

$(a)^{4}=a*a*a*a=(a*a*a)*a=c*a=e$

So,order of group$=4$

$\Rightarrow$Order of group$=$Order of generator$=4$