Algebra Theory?

Question

1. Set S and a binary operation * on S will be specified. In each case, determine whether * is associative, whether * has an identity element and, if the latter is the case, whether each element of S has an inverse (with regard to *), i.e whether S is a group:
(a) S=3Z, the set of all integers that are multiples of 3, * = ordinary addition of integers
(b) S=Z, a*b=a+b-1

2. A table defines a binary operation * on the set S={a,b,c,d}. Is (s,*) a group? If not why?

If someone could please give me full working out and explinations I would really appreciate it.

Thanks

Don

how do you get all the symbols..such as "for all"

Answer 1

1a: This is a group. Obviously associativity is satisfied, because * is just ordinary addition, which is associative. An identity element exists, namely 0 (since 0=3×0, and is thus a multiple of 3). And if n∈3Z, then n=3k for some k, which means that -n = 3(-k), so -n∈3Z -- thus, the inverse element of any element of 3Z is also in 3Z.

1b: This is also a group. Here we need to be more careful, since the operation is not just ordinary addition:

Associativity: We need to show that ∀a, b, c, (a*b)*c = a*(b*c). We do this by manually calculating each:

(a*b)*c = (a+b-1)*c = (a+b-1) + c - 1 = a+b+c - 2
a*(b*c) = a*(b+c-1) = a + (b+c-1) - 1 = a+b+c - 2

So associativity checks. Next, we note that 1 is an identity element of *, since ∀a:

a*1 = a+1-1 = a, and
1*a = 1+a-1 = a

Finally, we check for inverses. for a∈Z, let b = -a+2. Then we have that:

a*b = a + (-a+2) - 1 = 1, and
b*a = (-a+2) + a - 1 = 1

so b is an inverse of a, demonstrating that one exists.

2: It may or may not be a group, depending on the table. Certainly there are some tables which define a group operation on S, for instance:

...a b c d
-------------
a|a b c d
b|b c d a
c|c d a b
d|d a b c

And some which do not:

...a b c d
-------------
a|a a a a
b|a a a a
c|a a a a
d|a a a a

So you would need to see the table to know whether or not (S, *) is a group.

Edit: I input the symbols using the the quick unicode input tool from cardbox -- you can download it here: http://www.cardbox.com/quick.htm . All the symbols are of course, already available on your character map, this just makes writing them faster.