劉覺平群論期中考試及答案

1、The middle examination in Group TheorySchool of Physics and Technology, Wuhan University, Winter semester, 2012. 1. (1) What is the order of an element in the group?(2) Show that a group must be an Abelian group if the order of any element in the group, except for the identity , is two.Solution：(2)

2、Because for any group element , we haveorIn particular for , we have2. (1) What is the subgroup of a group? (2) What is the non-trivial subgroup of a group?(2) Prove a group cannot be expressed as the union of its two non-trivial subgroups. Proof: (3) Suppose that and are two non-trivial subgroups o

3、f a group ,and . Then it must leads and thus there must be , with and (1) and (2)so that Therefore or However, if , thenthis leads to a contradiction with (2). Similarly, , it leads to a contradiction with (1) as well. Altogether, we conclude that .3. (1) What is the concept for a homomorphism betwe

4、en two groups? (2) Suppose that is a homomorphism fromonto . Prove that ， ； ， where and are the units of the groups and .(3) Prove that the kernel of the homomorphism , i.e. , is a subgroup of .Solution:(2) From the above homomorphism and ， ， for we have, for and, for These imply that and (3) If , t

5、hen , Thereforenamely End. 4. Suppose a group G acting on a set . (1) Given any point in the set , prove that the subset of given byis a subgroup of , called the isotropy group of . (2) For two different points and in the same orbit under the action of . Prove that Proof: (1) We want to prove that t

6、he set is really a subgroup of G.At first, it is non-empty because which implies that Further, it is clear that if , i.e. , then , and if , i.e. and , then , .(2) 5. (1) Suppose that a group G act on two different sets M1 and M2 . What is the concept of a G-morphism from M1 to M2?(2) Let a group G a

7、cts on an arbitrary set , and acts on the set by conjugation Show that show that (3) Under the conditions given in (2), prove that the mapping f from to : is a G-morphism.Solution: (1) Let a group G acts on two different sets M1 and M2 . Given a mapping form M1 to M2 : (1)which is said to be equival

8、ent with respect to the actions of G, or f is a G-morphism if (2)i.e. G G (3)In other words, it does not matter whether we first apply a group element and then the mapping f , or first apply f and then the group element (4)Because of the arbitrariness of the choice of the point in and the element a in G . (2) Suppose that , the there are