Frobenius theorem on group actions: For each of the following groups, find a group where the given group is a different normal subgroup of the total. a) S3 b) A4 c) S4
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Frobenius theorem on group actions:
For each of the following groups, find a group where the given group is a different normal subgroup of the total.
a) S3
b) A4
c) S4
Argue your answer. Please be as clear as possible, showing and explaining all the steps, use definitios if necessary. Thank you a lot
Step by step
Solved in 4 steps
- Exercises 1. List all cyclic subgroups of the group in Example of section. Example 3. We shall take and obtain an explicit example of . In order to define an element of , we need to specify , , and . There are three possible choices for . Since is to be bijective, there are two choices for after has been designated, and then only one choice for . Hence there are different mappings in .Find subgroups H and K of the group S(A) in example 3 of section 3.1 such that HK is not a subgroup of S(A). From Example 3 of section 3.1: A=1,2,3 and S(A) is a set of all permutations defined on A.Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?
- Exercises 22. List all the distinct subgroups of each group in Exercise. Exercise 21. Suppose is a cyclic group of order. Determine the number of generators of for each value of and list all the distinct generators of . a. b. c. d. e. f.40. Find subgroups and of the group in example of the section such that the set defined in Exercise is not a subgroup of . From Example of section : andis a set of all permutations defined on . defined in Exercise :Exercises 19. Find cyclic subgroups of that have three different orders.
- 3. Consider the group under addition. List all the elements of the subgroup, and state its order.Find groups H and K such that the following conditions are satisfied: H is a normal subgroup of K. K is a normal subgroup of the octic group. H is not a normal subgroup of the octic group.11. Find all normal subgroups of the alternating group .
- Exercises 3. Find the order of each element of the group in Example of section. Example 3. We shall take and obtain an explicit example of . In order to define an element of , we need to specify , , and . There are three possible choices for . Since is to be bijective, there are two choices for after has been designated, and then only one choice for . Hence there are different mappings in .Let be a group of order 24. If is a subgroup of , what are all the possible orders of ?Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.