Test by Bikram | Mathematics | Test 2 | Question: 25

Question

  See the above table of  a,b,c,d. The total number of subgroups possible from the above diagram are _______.

Answer 1

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 .

Answer 2

{a},{a,c},{a,b,c,d}

Comments

how {a,b} can be subgroup?

b*b=c which is not closed under {a,b}

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yes it should be {a,c} and NOT {a,b} .

Thanks for correction , you can see my detailed answer above.

Answer 3

As per Lagrange's theorem, the order of a subgroup divides the order of the group.

Order/Cardinality of the group = 4.

So, possible subgroups can be of the order: 1, 2 and 4.

 

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Of order 1, we can only have the trivial subgroup that contains just the identity element.

Here, when a operates with a, b, c and d — it results in the same element.

=> a is the identity.

So, {a} is possible.

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Of order 2

Is {a,b} possible?

No. Because b*b gives c. Closure violated.

 

Is {a,c} possible?

Yes.

 

Is {a,d} possible?

No, because d*d gives c. Closure violated.

 

So, just {a,b} possible here.

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Of order 4 we have {a,b,c,d}

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Hence, three total subgroups are possible.