- #1

- 35

- 0

## Main Question or Discussion Point

Hello everybody,

I I have an exercise here that I'm really stuck with..

The "conjugacy part" in this book ('Algebra' by Dummit, Foote) is about

p-subgroups of G being subgroups of a conjugate of a Sylow p-subgroup

in G.

I tried some different approaches but can't get nowhere..

Using the "conjugacy part of the thm" for P intersect N has no

use since already P intersect N is a subgroup of P itself, so

we have to use it for another subgroup.

I also considered Q a sylow p-subgroup of N. Eventually I would

like to show that Q and (P intersect N) have the same cardinality.

Well, (P intersect N) is a p-subgroup of N and Q is a sylow p-subgrp of N

so that from "the conjugacy part of Sylow's thm" we have that

(P intersect N) is a subgroup of a conjugate of Q in G, that is,

(P intersect N) <= gQg^-1, some g in G.

That seems to be something but can't go more far..

Finally, the fact that N is normal smells like 2nd iso theorem..

From this, we deduce that (P intersect N) is normal sbgrp of P

and N is a normal sbgrp of NP.. Also

P / (P intersect N) is isomorphic to NP / N.

So we can play with the cardinalities.

But can't think of anything else :(

Any ideas or hints highly appreciated!

Thanks in advance!

PS. I wasn't sure if I should post this in the 'homework' section. This is an exercise

given in a first year graduate course in Algebra and I think that it shouldn't be put

with "calculus and beyond" questions..

EDIT: Actually the exercise has another part: "Deduce that PN/N is a sylow p-sbgrp of G/N".

Well, there is a chance that we don't need the normality of N to show the first part of the question (the thing that I asked)

I I have an exercise here that I'm really stuck with..

*Let P be in Syl_p(G) and assume N is a normal subgroup of G. Use the conjugacy part of Sylow's theorem to prove that P intesect N is a Sylow p-subgroup of N.*The "conjugacy part" in this book ('Algebra' by Dummit, Foote) is about

p-subgroups of G being subgroups of a conjugate of a Sylow p-subgroup

in G.

I tried some different approaches but can't get nowhere..

Using the "conjugacy part of the thm" for P intersect N has no

use since already P intersect N is a subgroup of P itself, so

we have to use it for another subgroup.

I also considered Q a sylow p-subgroup of N. Eventually I would

like to show that Q and (P intersect N) have the same cardinality.

Well, (P intersect N) is a p-subgroup of N and Q is a sylow p-subgrp of N

so that from "the conjugacy part of Sylow's thm" we have that

(P intersect N) is a subgroup of a conjugate of Q in G, that is,

(P intersect N) <= gQg^-1, some g in G.

That seems to be something but can't go more far..

Finally, the fact that N is normal smells like 2nd iso theorem..

From this, we deduce that (P intersect N) is normal sbgrp of P

and N is a normal sbgrp of NP.. Also

P / (P intersect N) is isomorphic to NP / N.

So we can play with the cardinalities.

But can't think of anything else :(

Any ideas or hints highly appreciated!

Thanks in advance!

PS. I wasn't sure if I should post this in the 'homework' section. This is an exercise

given in a first year graduate course in Algebra and I think that it shouldn't be put

with "calculus and beyond" questions..

EDIT: Actually the exercise has another part: "Deduce that PN/N is a sylow p-sbgrp of G/N".

Well, there is a chance that we don't need the normality of N to show the first part of the question (the thing that I asked)

Last edited: