2.1. FIRST RESULTS 91 2.1.7. Suppose that ' W G ! H is an isomorphism of groups. Show that G is abelian if, and only if, H is abelian. The following several exercises investigate groups with a small num-ber of elements by means of their multiplication tables. The requirements ea D a and ae D a for all a determine one row and one column of the multiplication table. The other constraint on the multiplication table that we know is that each row and each column must contain every group ele-ment exactly once. When the size of the group is small, these constraints sufﬁce to determine the possible tables. 2.1.8. Show that there is up to isomorphism only one group of order 2. Hint: Call the elements f e;a g . Show that there is only one possible mul-tiplication table. Since the row and the column labeled by e are known, there is only one entry of the table that is not known. But that entry is de-termined by the requirement that each row and column contain each group element. 2.1.9.
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