In: Advanced Math
(The “conjugation rewrite lemma”.) Let σ and τ be permutations.
(a) Show that if σ maps x to y then στ maps τ(x) to τ(y).
(b) Suppose that σ is a product of disjoint cycles. Show that στ has the same cycle structure as
σ; indeed, wherever (... x y ...) occurs in σ, (... τ(x) τ(y) ...) occurs in στ.
(a) Given that , we
have
(b) Suppose that
be a cycle in
. Thus, we have,
By part a) we have
which shows that
is a cycle in
. Thus, corresponding to every cycle
in
we get a cycle
in
, and conversely, for every cycle
in
we get a cycle
in
. Hence, if
is a product of cycle, so is
, and both has same cycle structure.