Group automorphism, Mathematics

(a) Find an example of groups G, H, K with K 2325_Group automorphism.png H and H 2325_Group automorphism.png G but K 1662_Group automorphism1.png G.

(b) A subgroup H of G is characteristic if σ(H) ⊆ H for every group automorphism σ of G. Show that every characteristic subgroup is normal.

(c) Show that if K ⊆ H ⊆ G, K is characteristic in H and H is characteristic in G then K is characteristic in G.

(d) Show that Dn(G) is normal in all of G not just in Dn-1(G).

(e) Conclude that G is solvable i it has a subnormal series in which each quotient is abelian and each group in the series is normal in all of G.

If G is a group, an automorphism of the form 1184_Group automorphism2.png is called an inner automorphism. An automorphism not of this form is called outer.

The aim of the following questions is to show that all the automorphisms of Sn are inner, except for S6 which does have an outer automorphism.

Posted Date: 2/26/2013 3:03:17 AM | Location : United States







Related Discussions:- Group automorphism, Assignment Help, Ask Question on Group automorphism, Get Answer, Expert's Help, Group automorphism Discussions

Write discussion on Group automorphism
Your posts are moderated
Related Questions

Find out the linear approximation for at x =8 .  Utilizes the linear approximation to approximate the value of  and Solution Since it is just the tangent line there


HOW MANY ZERO ARE THERE AT THE END OF 200

Probability of A is 85% Probability of B is 45% Probability A and B 56% What is the probability of not either A or B?

DEVELOPING PRE-NUMBER CONCEPTS :  Previously you have read how children acquire concepts. You know that, for children to grasp a concept, they must be given several opportunities

We know that one has to deal with numbers in day-to-day life irrespective of his inclination and field of work. Also one cannot refute the fact

i have this data 48 degree, 72 degree, 43.2degree, 24degree , 40.8degree on this make a pie chart

E1) Why don't you think of some activities for the same purpose now? E2) Suggest, in detail, another activity for helping a child grasp the algorithm for division. We come to

The sum of two consecutive even integers is the number 126. What are the integers? Two consecutive even integers are numbers in sequence, such as 4 and 6 or -30 and -32, that a