Consider the following groups

Consider the following groups.

Let G be the set of nonzero real numbers with respect to multiplication.

Let H be the set of matrices of the form (a 0) (0 b) where a, b are nonzero real numbers. The operation on H is matrix multiplication. You do not need to show that G and H are groups.

Are G and H isomorphic? Justify your answer.

Hint: Use Theorem 6.2 from Gallian

Let G be an abelian group of order 55 which contains elements a, b with ?a? =5, ?b? =11.

Prove that G is isomorphic to Z₅₅.

Hint: Look at ab.

Let G be the group of symmetries of R² with respect to the composition. Let K, l be position integers and let R be the rotation around the origin by K/l degrees.

Find the order of R in G. Justify your answer.

Hint: Use the formula for the order of $a^k$ if you know the order of $a$.