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Topic: Identity table questions (Read 321 times) |
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Mugwump101
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Identity table questions
« on: Jan 15th, 2007, 3:31pm » |
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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.
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Icarus
wu::riddles Moderator
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Re: Identity table questions
« Reply #1 on: Jan 15th, 2007, 6:36pm » |
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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 b, but try to find it yourself before looking. The same process also finds the value of the other expression: a.
FYI, the group you have here is officially called Z4.
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| « Last Edit: Jan 15th, 2007, 6:40pm by Icarus » |
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When their digits are reversed? " - Anonymous
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