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MATH 402A - Solutions for Homework Assignment 3

MATH 402A - Solutions for Homework Assignment 3. Problem 7, page 55: We wish to find C(a) for each a S3 . It is clear that C(i) = S3 . For any group G and any a G, it is clear that every power of a commutes with a and therefore (a) C(a) . Assume that a S3 and a 6= i. Then a has order 2 or 3. Thus, the subgroup (a) of S3 has order 2 or 3. Since (a) is a subgroup of C(a) and C(a) is a subgroup of S3 (as proved in class one day), Lagrange's theorem tells us that |(a)| divides |C(a)| and that |C(a)| divides |S3 | = 6.

MATH 402A - Solutions for Homework Assignment 3 Problem 7, page 55: We wish to find C(a) for each a ∈ S3. It is clear that C(i) = S3. For any group G and any a ∈ G, it is clear that every power of a commutes with a and therefore (a) ⊆ C(a) . Assume that a ∈ S3 and a 6= i. Then a has order 2 or 3. Thus, the subgroup (a) of S3 has order ...

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