SECTION 4.2 Basics of Group Theory 215
- Are the non-zero elements{ 1 , 2 , 3 , 4 , 5 }inZ 6 closed under mul-
tiplication? - Are the non-zero elements ofZp, wherepis a prime number, closed
under multiplication? - For any positive integern, setNn ={ 1 , 2 ,...,n}, and let P(Nn)
be the power set ofNn(see page 189). Show that for any integer
N with 0≤N ≤ 2 nthere exists subsetsA,B ⊆P(Nn) such that
(a) |A|=|B|=N,
(b) Ais closed under∩, and
(c) Bis closed under∪.(Hint: Use induction, together with the De Morgan laws.)4.2.3 Properties of binary operations
Ordinary addition and multiplication enjoy very desirable properties,
most notably, associativity and commutativity. Matrix multiplication
is also associative (though proving this takes a little work), but not
commutative. The vector cross product of vectors in 3-space is neither
associative nor is it commutative. (The cross product is “anticommu-
tative” in the sense that for vectorsu andv, u×v =−v×u. The
nonassociativity is called for in Exercise 2, below.) This motivates the
following general definition: letS be a set with a binary operation∗.
We recall that
∗isassociativeifs 1 ∗(s 2 ∗s 3 ) = (s 1 ∗s 2 )∗s 3 , for alls 1 , s 2 , s 3 ∈S;∗iscommutativeifs 1 ∗s 2 =s 2 ∗s 1 , for alls 1 , s 2 ∈S.Next, we say thateis anidentitywith respect to the binary oper-
ation∗if
e∗s=s∗e=s for all s∈S.