Abstract Algebra Answers

Describe (list the elements, give the identity and inverses) the cyclic

group generated (under multiplication) by [ 1 1 ]

[0 1] matrix

Prove that every cyclic group is abelian. give an example to show the converse is NOT true.

Prove or disprove: the set of diagonal n × n matrices with no zeroes on the diagonal is a subgroup of

GL(n,R)

Let D be an integral domain. Then prove that there exists a 

field F that contains a subring isomorphic to D.

Show that the lines

r·(4i+3j)−1=0, r·(−2i+j)−1=0, r·(i+2j)−1=0 ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃

are concurrent.

We showed that if S and T are isomorphic groups and e is the identity for S ,φ(e) gives the identity for T. Now show that if a and a′ are inverses in S, then φ (a) and φ (a′) are inverses in T

. (In other words, show that φ (a′) = (φ(a))'

Let G be a group with operation * and identity e. Prove that if

x * x = e for all x ∈ G, then G is abelian. Hint: consider

( a * b) * (a*b )

Let S be the set of all real numbers except −1. Define * on S by

a * b = a + b + ab

Show that〈 S,*〉is a group and solve 2 * x * 3 = 7 in S

.

Prove that a group has exactly one element with the property that

x * x = x (this is called an idempotent element)

Prove that if phi : S ---> T is an isomorphism of <S,*> with <T, #> and psi: T ---> U is an isomorphism of <T, #> with <U, diamond> then the composite function psi composed with phi is an isomorphism of <S, *> with <U, diamond>