Group Theory: Suppose that H and K are distinct subgroups of G of index 2. Prove that H∩K is a normal subg?
Suppose that H and K are distinct subgroups of G of index 2. Prove
that H∩K is a normal subgroup of G of index 4 and that G/(H∩K)
is not cyclic.
This one has confused me a bit...thanks for any help you can offer
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First show that all subgroups of index two are normal, then show the intersection of normal subgroups is normal.
I would say something like "let a be in H∩K", then show for that all g, gag^-1 ∈ H and gag^-1 ∈ K.
The second part is a wee bit tougher, and I'll answer it after I come back from a walk.
Given: G a team, H,ok subgroups with H?ok = {e}, G = HK ={hk | h in H, ok in ok}. think for any g in G that g = hk = h'ok'. then h'^-1h = ok'ok^-a million, whence h'^-1h (= ok'ok^-a million) is in H?ok. to that end h'^-1h = e, so h = h' (multiplying the two sides on the left via h'). from hk = h'ok' = hk', we've that ok = ok', besides (multiplying on the left via h^-a million).