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Could someone please help me with this question: Compute the Cayley table of the group $\mathbb{Z}^*_{10}$ and give an explicit definition for the different homomorphisms $\psi_1 ,$$\psi_2$$\quad$ $\mathbb{Z}^*_{10}$$\to$$\mathbb{Z}^*_{10}$.

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Where do we start? Do you know what the notation ${\bf Z}_{10}^*$ means? Do you know what a Cayley table of a group is? Do you know what a homomorphism is? Help us help you. – Gerry Myerson Feb 9 '13 at 5:02
Thank you, yes i understand what homomorphism is and for the notation $\mathbb{Z} ^*_{10}$ it is the multiplicative modulo 10 but i have a problem in computing this notation and what a Cayley table of a group is. – Bulou Duikoro Feb 9 '13 at 5:10
Well, I think "Cayley table" just means the multiplication table of the group. Now, in a multiplicative group, every element has to have a multiplicative inverse. Do you know what the identity element is in ${\bf Z}_{10}^*$? Can you figure out which elements have a multiplicative inverse? Then you can write down the multiplication table for those elements (under multiplication modulo $10$). By the way, if you want to be sure I'll see a comment addressed to me, you have to write @Gerry. – Gerry Myerson Feb 9 '13 at 5:26
@BulouDuikoro: In fact, the Cayley table tells you everything you could possibly want about a group, because it tells you exactly what the operation on the group is. You can use it to compute any product, you can use it to figure out inverses, and.... – Babak S. Feb 9 '13 at 5:39
You have to do better than that. You don't understand "multiplication table of the group"? You don't understand "multiplicative inverse"? You don't understand "identity element"? You don't understand "modulo $10$"? – Gerry Myerson Feb 9 '13 at 5:50

Here are simple steps you may follow. 1.List the non zero element of modular 10 which are 1,2,3,4,5,6,7,8,9 . 2.Determine the value of gdc(k,n)=1 i.e gcd(1,10)=1 , gcd(2,10)=2,gcd(3,10)=1, gcd(4,10)=2,gcd(5,10)=5,gcd(6,10)=2,gcd(7,10)=1,gcd(8,10)=2,gcd(9,10)=1. 3.The elements you need to form the Cayley table are 1,3,7,9

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