Mathematics 893 Problem Set 5

Suggested reading: Continue studying section 4.5, paying particular attention to the proof of the Sylow Theorems and to the discussion of groups of order pq and 30 (page 143). Exercise 26 on page 61 may be useful in solving one of the problems in part (b). Chapter 7, sections 1-4.

Questions due next Friday (October 21):

(a) Prove that any subgroup of index 2 is normal. (This might be proved somewhere in the text. Try to do it using only the definition of normal subgroup.)

(b) Page 146 #4, 5, 12, 16, 24, 33, 34

(c) Page 226 Prove parts (3) and (4) of Proposition 1.

(d) Page 231 #14 Notes: In my world rings normally have units. In fact, some texts that I am familiar with include a unit in the definition of ring. The language of categories might help keep things straight. There are (at least) two categories that one might consider. In the category of rings with unit, rings have units, and morphisms (ring homomorphisms) preserve the unit. In the category of rings without unit, rings need not have a unit element, and if a ring does have a unit, morphisms need not preserve it. We will usually be working in the category of rings with unit.

In the category of rings with unit, the kernel of a ring homomorphism is not a ring (but it is an ideal), so statements in the text like Proposition 5 (2), page 240, have to be read with some care. Similarly Theorem 8 (1), page 246, does not make sense in the category of rings with unit.