Question about a Theorem in Gallian's "Contemporary Abstract Algebra" I'm using this book as a reference for my Algebra course, and there's a lemma in the book that is really confusing me. It is on Page 102 of the Sixth Edition, for those who have the book. The lemma states: If [tex]\epsilon=\beta_1\beta_2\cdots\beta_r[/tex] where the [tex]\beta 's[/tex] are 2-cycles, then r is even. The author states that it is a special case of the theorem which says: if a permutation A can be expressed as a product of an even(odd) number of 2-cycles, then every decomp. of A into a product of 2-cyles must have an even(odd) number of 2-cycles. But doesn't the lemma state that every cycle can be written as a product of an even number of two cycles? I'm confused, and I'm not following the proof of the lemma. Any help would be appreciated.