Hello all,

I'm brand new to the forum and my name is Kalish. I'm from Russia. I have some questions on abstract algebra and differential geometry. Hope you guys will be able to help! Thanks in advance.

Problem statement: Let D_infinity denote the group of symmetries of a circle. Let SD_infinity denote the subgroup of D_infinity consisting of only the rotations. (By the way, SD_infinity is isomorphic to R/Z and to the complex unit circle.)

How can I find [D_infinity:SD_infinity]? (The index of SD_infinity in D_infinity). Is it 2, because the number of cosets is just the identity and the composition of rotations on the flips?

Also, how can I prove that D_infinity is not isomorphic to SD_infinity x {-1,1}? What group-theoretic property do they NOT share, such that I can separate the two? I'm stumped on this one.