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Self doubt Kenneth Rosen Algebraic structures

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In case of A having 4 elements {a,b,c,d} with A as Identity element.Number of groups possible is given to be 2. I want to know why this 

* a b c d
a a b c d
b b a d c
c c d b a
d d c a b

Is not a group in given case.

Also is there any formula that can give us number of groups possible when cardinality of set A is given?

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