Are these groups isomorphic

[From: ] [author: ] [Date: 11-10-29] [Hit: ]

which always has order one).so if we have a generator, it must be one of the other three.now ([0],[1]) + ([0],[1]) = ([0+0],......

clearly adding ([0],[0]) repeatedly will only yield ([0],[0]) (not surprising it IS the identity, which always has order one).

so if we have a generator, it must be one of the "other three".

now ([0],[1]) + ([0],[1]) = ([0+0],[1+1]) = ([0],[2]) = ([0],[0]) (2 is 0 mod 2)

so ([0],[1]) has order 2, thus it generates a group of order 2, so it doesn't generate B

(which has order 4).

([1],[0]) + ([1],[0]) = ([1+1],[0+0]) = ([2],[0]) = ([0],[0]) so this is also not a generator for B.

one more to go:

([1],[1]) + ([1],[1]) = ([1+1],[1+1]) = ([2],[2]) = ([0],[0]), no this element has order 2, as well.

we have NO generator, so B is not cyclic, so B is not isomorphic to G.

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Note that there are only two kinds of groups of order 4:
namely the cyclic group C4 and the dihedral group D4 (or also known as the Klein four group).
Each type is distinct (not isomorphic) to each other.

Group G and a are isomorphic to C4.
Note that each group has two generators
(i and -i for G; [1] and [3] for a)

However, group b is isomorphic to D4.
Note that each non-identity element has order 2
and the product of any two non-identity elements will yield the third one.
Such characteristics are found in D4 groups.

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