MAD 4301 Midterm: MAS4301 S15 Test 3
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Let g be a group and x g. prove: if x has order d and xk = e, then d | k. (3) (15 pts. ) Let g be a nite group and x g. prove: if h is a subgroup of. G, then the sets xh and h have the same number of elements. (4) (15 pts. ) Prove: if g is a nite abelian group of order n and x g, then xn = e. (5) (20 pts. ) List the cosets of h in g. (d) find a subgroup h in d4 of order 4. List the cosets of h in g. (6) (20 pts. ) Let r = mat2(z2) be the ring of 2-by-2 matrices with entries from the eld z2. Write 0 for the zero matrix, and i for the multiplicative identity matrix.
