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Exercises 6.5 Exercises

1.

Suppose that G is a finite group with an element g of order 5 and an element h of order 7. Why must |G|β‰₯35?

2.

Suppose that G is a finite group with 60 elements. What are the orders of possible subgroups of G?

3.

Prove or disprove: Every subgroup of the integers has finite index.

4.

Prove or disprove: Every subgroup of the integers has finite order.

5.

List the left and right cosets of the subgroups in each of the following.

  1. ⟨8⟩ in Z24

  2. ⟨3⟩ in U(8)

  3. 3Z in Z

  4. A4 in S4

  5. An in Sn

  6. D4 in S4

  7. T in Cβˆ—

  8. H={(1),(123),(132)} in S4

6.

Describe the left cosets of SL2(R) in GL2(R). What is the index of SL2(R) in GL2(R)?

7.

Verify Euler's Theorem for n=15 and a=4.

8.

Use Fermat's Little Theorem to show that if p=4n+3 is prime, there is no solution to the equation x2β‰‘βˆ’1(modp).

9.

Show that the integers have infinite index in the additive group of rational numbers.

10.

Show that the additive group of real numbers has infinite index in the additive group of the complex numbers.

11.

Let H be a subgroup of a group G and suppose that g1,g2∈G. Prove that the following conditions are equivalent.

  1. g1H=g2H

  2. Hg1βˆ’1=Hg2βˆ’1

  3. g1HβŠ‚g2H

  4. g2∈g1H

  5. g1βˆ’1g2∈H

12.

If ghgβˆ’1∈H for all g∈G and h∈H, show that right cosets are identical to left cosets. That is, show that gH=Hg for all g∈G.

14.

Suppose that gn=e. Show that the order of g divides n.

15.

The cycle structure of a permutation Οƒ is defined as the unordered list of the sizes of the cycles in the cycle decomposition Οƒ. For example, the permutation Οƒ=(12)(345)(78)(9) has cycle structure (2,3,2,1) which can also be written as (1,2,2,3).

Show that any two permutations Ξ±,β∈Sn have the same cycle structure if and only if there exists a permutation Ξ³ such that Ξ²=Ξ³Ξ±Ξ³βˆ’1. If Ξ²=Ξ³Ξ±Ξ³βˆ’1 for some γ∈Sn, then Ξ± and Ξ² are conjugate.

16.

If |G|=2n, prove that the number of elements of order 2 is odd. Use this result to show that G must contain a subgroup of order 2.

17.

Suppose that [G:H]=2. If a and b are not in H, show that ab∈H.

18.

If [G:H]=2, prove that gH=Hg.

19.

Let H and K be subgroups of a group G. Prove that gH∩gK is a coset of H∩K in G.

20.

Let H and K be subgroups of a group G. Define a relation ∼ on G by a∼b if there exists an h∈H and a k∈K such that hak=b. Show that this relation is an equivalence relation. The corresponding equivalence classes are called double cosets. Compute the double cosets of H={(1),(123),(132)} in A4.

21.

Let G be a cyclic group of order n. Show that there are exactly Ο•(n) generators for G.

22.

Let n=p1e1p2e2β‹―pkek, where p1,p2,…,pk are distinct primes. Prove that

Ο•(n)=n(1βˆ’1p1)(1βˆ’1p2)β‹―(1βˆ’1pk).

23.

Show that

n=βˆ‘d∣nΟ•(d)

for all positive integers n.