Parity.Commutators and Conjugation.

a$.$ Show that the for $g,$hinG,$[g,h{]}^{-1}=[h,g].$

b$.$ The T$-$perm $($given in part b$)$ swaps two corners and two edges on the top layer

as depicted below in Figure $1.$ What setup move$($s$)$ or conjugate $g$ can we apply

such that we are able to execute the modified swap depicted in Figure $2?$

Hint: If we take $h$ to be the T$-$perm, then we should have $gh{g}^{-1}$ result in the

swap given in Figure $2.$ Once again, using

alg.cubing.net could prove useful to

checking your solution. Type $"x2"$ in the setup section to start with the blue face

in front.

a$.$ Does the following cycle have even or odd parity? Write the cycle as a product of $2-$cycles.

b$.$ Identify the parity of the corners and the parity of the edges, separately after applying the T$-$perm algorithm below.

Hint: You can see how the cube looks after the algorithm on

alg.cubing.net.

R U R$\text{'}$ U$\text{'}$ R$\text{'}$ F R$2$ U$\text{'}$ R$\text{'}$ U$\text{'}$ R U R$\text{'}$ F'Parity And Composition. What is the parity of the product of an even parity cycle

and an odd parity cycle? How about the product of $2$ even parity cycles? How about $2$

odd parity cycles? Give a brief explanation.

Hint: We can write any even integer as $2n$ and any odd integer as $2m+1.$ Then consider

how the number of transpositions increases as we take the product of two cycles.