In Thomas Hungerford's "Abstract Algebra - An Introduction" he is giving following examples of products of cycles of permutations:
- $(567)^2=(576)$
- $(1234)=(12)(23)(34)$
- $(1243)(243)=(23)(34)(14)$
What I see in the first line is that in $(567)(567)$, the first cycle sends $5$ to $6$, and the second cycle sends $6$ to $7$, so that the result of the product is $5\rightarrow 7$ (and similarly we get $6\rightarrow 5$ and $7\rightarrow 6$. But if that is true, then it looks like the $3$ cycles on the right side of line $2$ sends $1\rightarrow2$, $2\rightarrow 3$ and $3\rightarrow 4$, so that the product would give $1\rightarrow 4$ and not $1\rightarrow 2$? Another doubt I have is that it seems that on the third line, $3$ is sent to two different numbers. Can someone help me understand how these products work?