Question about a Theorem in Gallian's Contemporary Abstract Algebra

[alligatorman]

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 $$\epsilon=\beta_1\beta_2\cdots\beta_r$$ where the $$\beta 's$$ 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.

 

[alligatorman]

I apologize. Turns out that $$\epsilon$$ is the identity permutation.

 

[JasonRox]

I apologize. Turns out that $$\epsilon$$ is the identity permutation.

Haha, I saw your post and had the book out (see my thread) and I was thinking the same thing at first glance, but then it's like... wait a second, e is the identity. It's not written like the typical e like the rest of the book, so I can understand the confusion.