Homework Statement
If J is a subgroup of G whose order is a power of a prime p, verify that J must be contained in a Sylow p-subgroup of G. The problem says to refer to a lemma that given an action by a subgroup H on its own left cosets, h(xH)=hxH, H is a normal subgroup iff every orbit of the...
Homework Statement
Write down all Sylow subgroups of A5 (alternating group). I have them for order 3 and order 5 subgroups, but am not sure about order 4 subgroups.
Homework Equations
Sylow Theorems etc
The Attempt at a Solution
I know the # of 4-subgroups is congruent to 1mod2...
H is a subgroup, so it includes e. For all h, h(xH)=e(xH)=xH. Left-multiplying by xi (x inverse) on both sides gives xihxH=H, so xihx belongs to H (for all x in G and h in H).
This is addressed to people who know complex analysis (hope this is the right section). Here's the Laurent theorem from my book for my later reference: Suppose a function f is analytic throughout an annular domain R1<|z-z0|<R2, centered at z0, and let C denote any positively oriented simple...