(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Question about a Theorem in Gallian's Contemporary Abstract Algebra

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