Show that there are only two distinct groups with four elements, as follows. Call the elements...

Show that there are only two distinct groups with four elements, as follows. Call the elements of the group e, a. b,c.

Let a denote a nonidentity element whose square is the identity. The row and column labeled by e are known. Show that the row labeled by a is determined by the requirement that each group element must appear exactly once in each row and column; similarly, the column labeled by a is determined. There are now four table entries left to determine. Show that there are exactly two possible ways to complete the multiplication table that are consistent with the constraints on multiplication tables. Show that these two ways of completing the table yield the multiplication tables of the two groups with four elements that we have already encountered.

Expert Solution

Colby Messinger answered 1 year ago

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