Q: 54 The groups An and Sn are not solvable for all n ≥ 5.
A: Solution:-
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A: As per our guidelines we are suppose to answer only one ques. Answer of question 1 is as follows:
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Q: How many subgroups of the group G = (Z30, +30)? O a. 4 O b. 7 O c. 5 O d. 6 O e. 8
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A: Given groups are,
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Q: (b) Complete the following character table of a group of order 12: 1 3 4 X1 X2 X3 4,
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Find the order of each of the following elements in the multiplicative group of units Up.
a. [2] for p = 13
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- Exercises 13. For each of the following values of, find all subgroups of the group described in Exercise, addition and state their order. a. b. c. d. e. f.Let a,b,c, and d be elements of a group G. Find an expression for (abcd)1 in terms of a1,b1,c1, and d1.6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.
- 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=19Let 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?9. Find all elements in each of the following groups such that . under addition. under multiplication.
- If a is an element of order m in a group G and ak=e, prove that m divides k.42. For an arbitrary set , the power set was defined in Section by , and addition in was defined by Prove that is a group with respect to this operation of addition. If has distinct elements, state the order of .12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.
- Exercises 10. For each of the following values of, find all subgroups of the cyclic group under addition and state their order. a. b. c. d. e. f.Find all subgroups of the octic group D4.Label each of the following statements as either true or false. The order of the identity element in any group is 1.