Question

The Klein 4-group is the order 4 group V={e,a,b,c} where (a) Show that V≅Z2×Z2. (b) Show that if ∣G∣=4 then either G is cyclic or G≅V.

Answer 1

The student's questions pertain to demonstrating that the Klein 4-group is isomorphic to Z2×Z2, and that any group of order 4 is either cyclic or isomorphic to the Klein 4-group.

Step-by-step explanation:

The student's question involves demonstrating two properties of group theory within abstract algebra:

1. Show that the Klein 4-group V is isomorphic to Z2×Z2, where V contains four elements {e,a,b,c}.
2. Show that if a group G has order 4, then G is either cyclic or isomorphic to the Klein 4-group V.

To show (a), note that V consists of four elements that operate under the group operation and that no element other than the identity e has order greater than 2. Hence, each non-identity element squares to the identity (aa=e, bb=e, cc=e), and the multiplication table of V mimics that of the direct product Z2×Z2.

To address (b), any group G of order 4 must abide by Lagrange's theorem, which implies each element has an order that divides the group order. If an element of order 4 exists, G is cyclic. Otherwise, no such element exists and, similar to the Klein 4-group, must have a structure where each element is of order 2, which by definition makes it isomorphic to V.