a.
To find the order of
a.
Explanation of Solution
Given:
The symmetric group on four elements.
Calculation:
Let us consider that
The order of symmetric group of
Therefore, the order of
b.
To compute the products
b.
Explanation of Solution
Let us consider that
Now, finding the product as follows-
Therefore,
Similarly,
c.
To compute the products
c.
Explanation of Solution
Let us consider that
Now, finding the product as follows-
Therefore,
Similarly,
d.
To find the inverses of
d.
Explanation of Solution
The objective is to find
So the inverse of
Similarly the inverse of
Similarly the inverse of
e.
To show that
e.
Explanation of Solution
For showing
So let us consider
Let us consider that
Now, finding the product as follows-
Since
Hence
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Chapter 6 Solutions
A Transition to Advanced Mathematics
- The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is. A4=D4.arrow_forwardExercises 10. Find an isomorphism from the multiplicative group to the group with multiplication table in Figure . This group is known as the Klein four group. Figure Sec. 16. a. Prove that each of the following sets is a subgroup of , the general linear group of order over . Sec. 3. Let be the Klein four group with its multiplication table given in Figure . Figure Sec. 17. Show that a group of order either is cyclic or is isomorphic to the Klein four group . Sec. 16. Repeat Exercise with the quaternion group , the Klein four group , and defined byarrow_forwardFind the order of each of the following elements in the multiplicative group of units . for for for forarrow_forward
- Use mathematical induction to prove that if a1,a2,...,an are elements of a group G, then (a1a2...an)1=an1an11...a21a11. (This is the general form of the reverse order law for inverses.)arrow_forwardProve or disprove that the set of all diagonal matrices in Mn() forms a group with respect to addition.arrow_forwardFind two groups of order 6 that are not isomorphic.arrow_forward
- 19. a. Show that is isomorphic to , where the group operation in each of , and is addition. b. Show that is isomorphic to , where all group operations are addition.arrow_forward2. Show that is a normal subgroup of the multiplicative group of invertible matrices in .arrow_forwardTrue or False Label each of the following statements as either true or false. In a Cayley table for a group, each element appears exactly once in each row.arrow_forward
- For each of the following values of n, find all distinct generators of the group Un described in Exercise 11. a. n=7 b. n=5 c. n=11 d. n=13 e. n=17 f. n=19arrow_forward15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,