If G is a group with subgroups A and B, then how do you define "x ~y" if and only if there exists a in A and b in B such that x = ayb?

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- Nov 30th 2011, 10:57 AMnep977define x ~ y
If G is a group with subgroups A and B, then how do you define "x ~y" if and only if there exists a in A and b in B such that x = ayb?

- Nov 30th 2011, 12:06 PMDevenoRe: define x ~ y
that IS the definition.

i suspect what you want to do is show "~" is an equivalence relation; that is, show ~ is reflexive, symmetric and transitive. - Nov 30th 2011, 12:31 PMnep977Re: define x ~ y
ok so

R is an equivalence relation

Its reflexive: y ~ y

Symmetry: x ~ y then y ~ x

is it like this? i don't get the part "if and only if there exists a in A and b in B such that x = ayb"

what do i do with x = ayb - Nov 30th 2011, 12:39 PMDevenoRe: define x ~ y
to prove that ~ is reflexive, you need to prove that for ANY x in G, you can find a in A and b in B such that x = axb.

you can't just say that x~x, because you don't KNOW that, unless you can actually produce the elements a and b. - Nov 30th 2011, 01:10 PMnep977Re: define x ~ y
ok got it. Thank you