Prove or disprove: If G = (V; E) is an undirected graph where
every vertex has degree at least 4 and u is in V , then there are
at least 64 distinct paths in G that start at u.
Let (G,+) be an abelian group and U a subgroup of G. Prove that
G is the direct product of U and V (where V a subgroup of G) if
only if there is a homomorphism f : G → U with f|U =
IdU
Using Kurosch's subgroup theorem for free proucts,prove that
every finite subgroup of the free product of finite groups is
isomorphic to a subgroup of some free factor.
1. Prove or disprove: if f : R → R is injective and g : R → R is
surjective then f ◦ g : R → R is bijective.
2. Suppose n and k are two positive integers. Pick a uniformly
random lattice path from (0, 0) to (n, k). What is the probability
that the first step is ‘up’?