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        riddles >> general problem-solving / chatting / whatever >> Identity table questions 
        
(Message started by: Mugwump101 on Jan 15^th, 2007, 3:31pm)

Title: Identity table questions
Post by Mugwump101 on Jan 15^th, 2007, 3:31pm

For the group given in the chart at the right find the identity and inverse of each element. As well as evaluate (a@b)@c and (d@d)@a.
(There's a chart....

@ I a I b I c I d I
a I d I a I b I c I
b I a I b I c I d I
c I b I c I d I a I
d I c I d I a I b I

I understand that the identity is b because it reveals the set but I don't understand how to do the rest.

Title: Re: Identity table questions
Post by Icarus on Jan 15^th, 2007, 6:36pm

The identity of a group G is the element e such that e@x = x@e = x for every element x. You are correct that b is the element with this property: b times anything is that thing.

The inverse of an element x is the unique element y such that x@y = y@x = e (the identity). In this case e = b, so for each element of the set, we need to find another element that "multiplies" by it to give b: Note that a@c = c@a = b, d@d = b, and of course b@b = b. So
a^-1 = c,
b^-1 = b,
c^-1 = a,
d^-1 = d.

As for the last, what is a@b? If a@b = x, then (a@b)@c = x@c. So all you have to do is find the value of a@b - which you can read straight from the chart - , then take that value "times" c, and look up that product. The final result is [hide]b[/hide], but try to find it yourself before looking. The same process also finds the value of the other expression: [hide]a[/hide].

FYI, the group you have here is officially called Z₄.