3.1 Examples and Basic Properties
1.
a. Not an additive group; only 0 has an inverse.
c. It is an additive abelian group (as in Example 4), unity is 
, multiplication is associative (it is function composition). But h 
(f + g) = h f + h g fails: h (f + g)(x) = h[f(x) + g(x)] while
, multiplication is associative (it is function composition). But h (f + g) = h f + h g fails: h (f + g)(x) = h[f(x) + g(x)] while
These may not be equal. Note that (f + g) h = f h + g h does hold.
h = f h + g h does hold.
3.
a. Clearly 
and 
. If and are in S then ...
and . If and are in S then ...Get Introduction to Abstract Algebra, Solutions Manual, 4th Edition now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
