2.1. FIRST RESULTS912.1.7.Suppose that'WG!His an isomorphism of groups. Show thatGis abelian if, and only if,His abelian.The following several exercises investigate groups with a small num-ber of elements by means of their multiplication tables. The requirementseaDaandaeDafor alladetermine one row and one column of themultiplication table. The other constraint on the multiplication table thatwe 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 constraintssuffice to determine the possible tables.2.1.8.Show that there is up to isomorphism only one group of order 2.Hint:Call the elementsfe; ag. Show that there is only one possible mul-tiplication table. Since the row and the column labeled byeare 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 groupelement.
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