Homework Nine Part I (of II)
(1) Let S be a finite set with n elements. How many different
permutations of S are there? Prove your answer.
(2) Describe a group structure on the set of moves of a Rubik’s
cube. If you are unsure what these moves are, you might find
the first few seconds of this video helpful. Your description
should include your chosen identity element, chosen binary operation and a brief explanation for why the group axioms hold.
(3) Let G be a finite group (i.e. a group with finitely many elements) with binary operation ? and identity eG . Prove that for
every g ∈ G there is some n ∈ N such that
(†)
g ? g ? g ? . . . ? g = eG .
n times
(4)
(5)
(6)
(7)
It might be helpful to figure this out concretely for your favourite
permutation group before tackling the case of an abstract group.
What does (†) mean in the context of the Rubik’s cube?
Let Γ and Ξ be graphs and suppose that there is a surjective
graph homomorphism ϕ : Γ → Ξ. Prove that if Γ is connected
then Ξ is also connected.
Prove or disprove: there is a graph homomorphism from a
square graph
to a graph consisting of a single edge
(the
answer is not what you initially think it is).
Prove that a graph whose vertices and edges can be drawn as
such
is not isomorphic to a graph whose vertices and
(here, as in the previous
edges can be drawn as such
question, it might be helpful to label the vertices before trying
to write anything down).
Find all the automorphisms of your favourite graph. In other
words, describe its automorphism group.
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