Question

In: Advanced Math

How many subgroups of order 9 and 49 may there be in a Group of order...

How many subgroups of order 9 and 49 may there be in a Group of order 441

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Expert Solution

We will Sylow third theorem to find number of subgroups of order 9 and 49 in a group of order 441 which states that ,

Sylow third theorem : Supppse be a group with where be a prime does not divides If denotes number of subgroup of order then ,

divides and

Now , and 3 does not divides 49 .

So if denotes number of subgroup of order 9 then by Sylow third theorem ,

and

Now 1, 7 , 49 are all satisfies the equation .

Hence possible number of subgroup of order are 1 , 7 , 49 .

Also , and 7 does not divides 9 so by Sylow third theorem if denotes number of subgroup of order 7 then ,

and

Now 1 satisfies but 3 , 9 does not divides .

Hence number of subgroup of order 49 is 1.

If you have any doubt or need more clarification at any step please comment.

Colby Messinger answered 1 year ago

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