Subgroups

Can someone please explain to me why S6 has no subgroups of order 14. I know it's going to involve Lagrange's theorem I'm just not sure how to find the size the group order 14 would be for comparison first. Thank you

Reply 1

Cryptokyo

14

Original post by MissMathsxo

Can someone please explain to me why S6 has no subgroups of order 14. I know it's going to involve Lagrange's theorem I'm just not sure how to find the size the group order 14 would be for comparison first. Thank you

Lagrange's theorem is that "if H is subgroup of G then the order of H divides the order of G". The contrapositive formulation of this is that "if the order of H does not divide the order of G then H is not a subgroup of G".

The order of S6 is 6!=720. What can we say about subgroups of order 14 using the contrapositive?

Reply 2

MissMathsxo

OP

12

Original post by Cryptokyo

Lagrange's theorem is that "if H is subgroup of G then the order of H divides the order of G". The contrapositive formulation of this is that "if the order of H does not divide the order of G then H is not a subgroup of G".

The order of S6 is 6!=720. What can we say about subgroups of order 14 using the contrapositive?

Yeah I got this part and I know that the subgroup of order 14 must divide G but I don't know how to show that it doesn't as I don't know how to find the size of the subgroup. Hope this makes sense☺️

Reply 3

Cryptokyo

14

Original post by MissMathsxo

Yeah I got this part and I know that the subgroup of order 14 must divide G but I don't know how to show that it doesn't as I don't know how to find the size of the subgroup. Hope this makes sense☺️

You are told that the order (size) of the possible subgroup in the question is 14. 14 does not divide 720 so S6 cannot have any subgroups of order 14 by Lagrange's theorem.

Or have I misunderstood?

Reply 4

MissMathsxo

OP

12

Original post by Cryptokyo

You are told that the order (size) of the possible subgroup in the question is 14. 14 does not divide 720 so S6 cannot have any subgroups of order 14 by Lagrange's theorem.

Or have I misunderstood?

This is what I thought the answer would be but the mark scheme confused me when it said to use 7 doesn't divide 720. Is this just because if 7 doesn't divide it 14 won't?

Reply 5

DFranklin

18

Original post by MissMathsxo

This is what I thought the answer would be but the mark scheme confused me when it said to use 7 doesn't divide 720. Is this just because if 7 doesn't divide it 14 won't?

Yes. (I think saying 14 doesn't divide it is a better answer, TBH).

Original post by MissMathsxo

This is what I thought the answer would be but the mark scheme confused me when it said to use 7 doesn't divide 720. Is this just because if 7 doesn't divide it 14 won't?

Yes...

If a number is divisible by 14 then it is divisible by 2 and 7.

The number 720 is div by 2, but not by 7, so its not div by 14.

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