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0. This exercise shows that there are two nonisomorphic group structures on a set of 4 elements. Let the set be {e, a, b, c}, with e the identity element for the group operation. A group table would then have to start in the manner shown in Table 2.29 D. The square indicated by the question mark cannot be filled in with a. It must be filled in either with the identity element e or with an element different from both e and a. In this latter case, it is no loss of generality to assume that this element is b. If this square is filled in with e, the table can then be completed in two ways to give a group. Find these two tables. (You need not check the associative law.) If this square is filled in with b, then the table Zoom Ou can only be completed in one way to give a group. Find this table. (Again, you need not check the associative law.) Of the three tables you now have, two give isomorphic groups. Determine which two tables these are, and give the one-to-one onto relabeling function which is an isomorphism. 2.29 Table e a e e a a a ? a. Are all groups of 4 elements commutative? b. Find a way to relabel the four matrices {C so the matrix multiplication table is identical to one you constructed. This shows that the table you constructed defines an associative operation and therefore gives a group. c. Show that for a particular value of n, the group elements given in Exercise 14 D can be relabeled so their group table is identical to one you constructed. This implies the operation in the table is also associative.

0. This exercise shows that there are two nonisomorphic group structures on a set of 4 elements. Let the set be {e, a, b, c}, with e the identity element for the group operation. A group table would then have to start in the manner shown in Table 2.29 D. The square indicated by the question mark cannot be filled in with a. It must be filled in either with the identity element e or with an element different from both e and a. In this latter case, it is no loss of generality to assume that this element is b. If this square is filled in with e, the table can then be completed in two ways to give a group. Find these two tables. (You need not check the associative law.) If this square is filled in with b, then the table Zoom Ou can only be completed in one way to give a group. Find this table. (Again, you need not check the associative law.) Of the three tables you now have, two give isomorphic groups. Determine which two tables these are, and give the one-to-one onto relabeling function which is an isomorphism. 2.29 Table e a e e a a a ? a. Are all groups of 4 elements commutative? b. Find a way to relabel the four matrices {C so the matrix multiplication table is identical to one you constructed. This shows that the table you constructed defines an associative operation and therefore gives a group. c. Show that for a particular value of n, the group elements given in Exercise 14 D can be relabeled so their group table is identical to one you constructed. This implies the operation in the table is also associative.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.1: Definition Of A Group
Problem 16E: In Exercises 15 and 16, the given table defines an operation of multiplication on the set S={...
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How do we classify the tables？Is there any rule，I awlays have trouble decideing whether it should be filled with b or c.
20. This exercise shows that there are two nonisomorphic group structures on a set of 4 elements. Let the set be {e, a, b, c}, with e the identity element for the group operation. A group table would then have to start in the manner shown in Table 2.29 D. The square indicated by the question mark cannot be filled in with a. It must be filled in either with the identity element e or with an element different from both e and a. In this latter case, it is no loss of generality to assume that this element is b. If this square is filled in with e, the table can then be completed in two ways to give a group. Find these two tables. (You need not check the associative law.) If this square is filled in with b, then the table can only be completed in one way to give a group. Find this table. (Again, you need not check the associative law.) Of the three tables you now have, two Zoom Out give isomorphic groups. Determine which two tables these are, and give the one-to-one onto relabeling function which is an isomorphism. 2.29 Table e a e e a a a ? b. a. Are all groups of 4 elements commutative? b. Find a way to relabel the four matrices so the matrix multiplication table is identical to one you constructed. This shows that the table you constructed defines an associative operation and therefore gives a group. c. Show that for a particular value of n, the group elements given in Exercise 14 D can be relabeled so their group table is identical to one you constructed. This implies the operation in the table is also associative.
Transcribed Image Text:20. This exercise shows that there are two nonisomorphic group structures on a set of 4 elements. Let the set be {e, a, b, c}, with e the identity element for the group operation. A group table would then have to start in the manner shown in Table 2.29 D. The square indicated by the question mark cannot be filled in with a. It must be filled in either with the identity element e or with an element different from both e and a. In this latter case, it is no loss of generality to assume that this element is b. If this square is filled in with e, the table can then be completed in two ways to give a group. Find these two tables. (You need not check the associative law.) If this square is filled in with b, then the table can only be completed in one way to give a group. Find this table. (Again, you need not check the associative law.) Of the three tables you now have, two Zoom Out give isomorphic groups. Determine which two tables these are, and give the one-to-one onto relabeling function which is an isomorphism. 2.29 Table e a e e a a a ? b. a. Are all groups of 4 elements commutative? b. Find a way to relabel the four matrices so the matrix multiplication table is identical to one you constructed. This shows that the table you constructed defines an associative operation and therefore gives a group. c. Show that for a particular value of n, the group elements given in Exercise 14 D can be relabeled so their group table is identical to one you constructed. This implies the operation in the table is also associative.
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