Question

For a rectangle

-describe the type of groups of a rectangle

-describe the orders of the groups

-describe the structure of the groups

-describe the elements of the groups (make sure to name all the elements and describe them as a group of permutations on the vertices)

- describe each group as subgroups of permutation groups

-describe all possible orders, types and generators for each subgroup of the group

-are any of these groups cyclic and or abelian?

-are any of these subgroups cyclic and or abelian?

-are there subgroups of every possible order?

-which subgroups are isomorphic and how do you know?

Answer 1

We can make the group D4, dihedral group of order 8 by 90o rotation with respect to the center of mass and 1800 rotation with respect to a fixed diagonal of a rectangle.

We consider two elements a and b where a is described by 900 rotation of the rectangle with respect to the center of mass and b is described by 1800 rotation with respect to a fixed diagonal.

Thus we get that the order of a is 4 and the order of b is 2. These a and b together make the dihedral group D4 i.e., D4 = <a,b>

A=<a> , B=<b> , C=<a2> are three subgroup of D4. .Clearly A, B, C are the cyclic subgroup of D4 and hence they are commutative. B*C is a non-cyclic commutative subgroup of D4. D4 is the only non-commutative subgroup of D4 because D4 is the only subgroup of D4 of order 8. All other subgroups must be of the order of 1,2 or 4 and hence commutative.

2,4 8 are the multiples of 8. B, A, D4 are the subgroup of order 2,4,8 respectively

Here B and C are of order 2 and hence they are isomorphic. A and B*C are of order 4. But A is isomorphic to Z4 and B*C is isomorphic to K4. So A and B*C are not isomorphic.