M417 Homework 6 Spring 2004
Instructions: Solutions are due Fri., March 12.
- A digraph (i.e., directed graph) is a set
of vertices, some of which can be connected by an arrow
(i.e., a directed edge). For example, we can associate to
each group G its subgroup digraph, in which each subgroup H of G is
represented by a vertex vH, and there is an arrow from
a vertex vH to a vertex vH' exactly when
H' properly contains H. A directed path
(of length r) in a digraph is a sequence v0,..., vr
of vertices such that for each 1 <= i <= r there is an arrow
from vi-1 to vi.
every directed path in the subgroup digraph of a cyclic group of order N
has length at most log2N.
- Let g and x be in Sn. Assume that x =
(a1 ... ar)
is an r-cycle. Show that gxg-1
= (g(a1) ... g(ar)).
- Find the centralizer of (1234) in S4. [Hint:
- Let n and N be positive integers.
- (a) If f: Zn -> ZN is
a homomorphism of groups and m = f(1),
show that N divides mn, and that f(x) = mx mod N, for
all x in Zn.
- (b) Conversely, if m is a positive integer
such that N divides mn, show that
f(x)= mx mod N defines a homomorphism
f: Zn -> ZN.
- Let f: G -> H be a homomorphism of groups.
If G is finite, show that |f(G)||ker f|=|G|.