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#1 2007-03-20 09:44:49

deepz
Member
Registered: 2007-03-20
Posts: 10

need some help with algebra

Hi I'm stuck on 2 questions on algebra.

1)Let G be a group with 4 elements {e,a,b,c} in which e is the identity and a^2=b. Write down the Cayley table of G, explaining your working.

2)G is the set of all rotations about the origin of the real Euclidean plane, the operation is the composition of the mappings. Is this a group?Justify your answer

I'd appreciate any help, thanks

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#2 2007-03-20 09:53:26

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: need some help with algebra

1.  This is probably "cheating" but there are only two groups of order 4, Z4 and Z2xZ2.  As it turns out, for all elements g in Z2xZ2, g^2 = e, the identity.  So it must be Z4.  I say cheating because you probably don't understand much other than the conclusion.

So lets draw the table instead:

__|_e_|_a_|_b_|_c_
e  |  e  | a  |  b |  c
a  |  a  | b   |      |
b  |  b  |     |      |
c  |  c  |     |      |

If ab = e, then ac = c, so a is the identity.  We know that a is not the identity, so ab = c.  Thus, ac = e.

It shouldn't be too hard from here on out.

2. Is there an identity rotation?  Are two rotations a rotation?  For every rotation, is there an inverse?  Are rotations associative?  (all mappings are associative)


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#3 2007-03-20 10:53:02

deepz
Member
Registered: 2007-03-20
Posts: 10

Re: need some help with algebra

thanks for the help!

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