Řešitelský seminář, 28.2.2017
Problem 1. Evaluate
lim  /   ex dx.
Jo
Problem 2. We say that a group (G, •) has the property (P), if for any automorphism f of G exist two automorphisms g and h of G such that f(x) = g(x) ■ h(x), for any x € G. Prove that:
1. Any group with the property (P) is commutative,
2. Any unite abelian group of odd order has the property (P),
3. No unite group of order 4n + 2, n € N, has the property (P).
Problem 3. Let A and B be two n by n matricies with real entries such that AB2 = A — B.
1. Prove that In + B is a nonsingular matrix,
2. Prove that AB = BA.
Problem 4. Let G be a group of order n and let e be the identity element. Find all functions f : G —> -/V* such that
1. f(x) = 1 iffx = e.
2. f(xk) = f(x)/(f(x), k), for all positive divisors k of n.
Domácí úloha
Problem 5. Let G be a group of order 2p, where p is an odd prime. Assume that G has a normal subgroup of order 2. Prove that G is cyclic.