Instructions: Please double space your answer, leave plenty of space between problems and staple your answer sheet a together. Please indicate which two p problems you would like graded in detail. As described in class, I will grade these two problems in detail out of fine points each, and will award a maximum of another five points based on a quick survey of the remaining problems. Remark: In a pure mathematical course such as this, you are expected to write clearly , and explain your arguments carefully in detail. There is no place in mathematical for sloppy argumentations. In a group G, If an element g has odd order n, show that g = (gZ)k for some integer k. Prove that a finite group G Of even order contains an element of order 2. (See the text, section 1,1, for a hint,) Let S be a finite set, and f : s rightarrow a function. If f is a subjective map, show that S is injective. If f is an injective map, show that S is subjective. Give an example of an infinite see S and a map f : S rightarrow S such that f is subjective but not injective. Give an example of an infinite set S and a map f : S rightarrow S such that f is injective but no subjective. If G is a group and a G, define the centralizer of a in G,C(a), to be the set {g G | ga = ag}. Show that G(a) is a subgroup of G. Show that G(a) contains at least two elements as long as|G| 2. If G is abelian, show that if a G has finite order and b G has finite order, then also has finite order. Show by contrast that in the group of invertible 2times2 matrices with entries in has order has order 3, but has infinite order.
Show transcribed image text Instructions: Please double space your answer, leave plenty of space between problems and staple your answer sheet a together. Please indicate which two p problems you would like graded in detail. As described in class, I will grade these two problems in detail out of fine points each, and will award a maximum of another five points based on a quick survey of the remaining problems. Remark: In a pure mathematical course such as this, you are expected to write clearly , and explain your arguments carefully in detail. There is no place in mathematical for sloppy argumentations. In a group G, If an element g has odd order n, show that g = (gZ)k for some integer k. Prove that a finite group G Of even order contains an element of order 2. (See the text, section 1,1, for a hint,) Let S be a finite set, and f : s rightarrow a function. If f is a subjective map, show that S is injective. If f is an injective map, show that S is subjective. Give an example of an infinite see S and a map f : S rightarrow S such that f is subjective but not injective. Give an example of an infinite set S and a map f : S rightarrow S such that f is injective but no subjective. If G is a group and a G, define the centralizer of a in G,C(a), to be the set {g G | ga = ag}. Show that G(a) is a subgroup of G. Show that G(a) contains at least two elements as long as|G| 2. If G is abelian, show that if a G has finite order and b G has finite order, then also has finite order. Show by contrast that in the group of invertible 2times2 matrices with entries in has order has order 3, but has infinite order.