1)prove that every finite subgroup of Q/Z is cyclic.
1) G={m1/n1 + Z,..., mk/nk + Z} finite simple of Q/Z where gcd(mi,ni)=1, set n=lcm(n1,...nk)
i don't know how to show that G is <1/n + Z>
2) invertible :b/a 11 doesnt divide 1 irred: 11k/b
3) Z/10Z ={(1),...(9)}
(2)=(4)
inver: (1)^(-1)=1, (2)^(-1)=8, (3)^(-1)=7, (4)^(-1)=4,(5)^(-1)=5,(6)^(-1)=6,(7)^(-1)=3,(8)^(-1)=2,(9)^(-1)=9
Let be a finite group. Let us agree to write elements in as .
. Now prove the following:
1)
2) From the above it follows that
3) If there's an element s.t. , then we get a contradiction to the maximality of n
4) It follows that
Tonio
Ps. BTW, from the above it almost follows, adding some minor details, that has a unique (cyclic, of course) subgroup of order for every