There are some properties regarding subgroup:
- Subgroup order should divide order of a group .
- Zero order group is not possible. so here order 1 subgroup is possible as (4/1) is possible , order 2 subgroup possible, order 3 subgroup Not possible and order 4 subgroup possible.
- And therefore only 1,2,4 order subgroup possible.
- Each case we need to check closure for inverse element and inclusion of identity element.
so {a} order 1 , order 2 {a,b} {a,c} {a,d} and order 4 subgroup {a,b,c,d} these are only possible in this diagram .
now a * a = a
for {a,b} --> a*b = b , b*a = b , a * a =a , b *b = c which is not closed under {a,b} so {a,b} subgroup is not satisfy the condition.
for {a,c} --> a*c=c, c*a= c, a*a = a , c*c = a so it satisfy the condition .
for {a,d} --> a*d =d , d*a= d , a*a= a , d*d = c which is not closed under {a,d} , so subgroup {a,d} is not possible .
and {a,b,c,d} a*b = b , a*c=c, a*d=d , a*a =a , b*b=c, c*c=a, d*d=c so subgroup {a,b,c,d} is possible .
Hence there are 3 subgroups possible {a} , {a,c} and {a,b,c,d} .
Answer is 3 .