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Homework answers / question archive / 5

5.2. Prove that the intersection KN H of subgroups of a group G is a subgroup of H, and

that if K is anormal subgroup of G, then K N H is a normal subgroup of H.

§.6. Determine the center of GL, (R).

Hint: You are asked to determine the invertible matrices A that commute with every

invertible matrix B. Do not test with a general matrix B. Test with elementary matrices.

6.8. Prove that the map A ~ (A')7! is an automorphism of GL, (R).

6.10. Find all automorphisms of .

(a) a cyclic group of order 10, (b) the symmetric group S3.