Group Theory Homeworks

Text: Groups and Symmetry (Undergraduate Texts in Mathematics) by M. A. Armstrong

Homework 1: Ch 2: 2.2, 2.8* Ch 3: 3.1(i), (ii), 3.2, 3.3, 3.10* (Number Theory example) Ch 4: 4.2, 4.4, 4.5, 4.6, 4.8* Ch 5: 5.5, 5.7, 5.10, Ch 6: 6.1, 6.3, 6.11

Homework 2:

Ch 12: 12.1, 12.2, 12.7 Ch 11: 11.3, 11.4, 11.12

Homework 3:

1. Find a bijection between the natural numbers (include 0) and even integers. Also find a bijection between the natural numbers (include 0) and odd integers. Don't forget to prove they are bijections.

2. Prove that the function f(x)=x^2 is not injective on the reals. Also prove that it is not surjective.

3. Label the square of the dihedral group D8 with 1,2,3,4 clockwise with 1 in the upper left corner (like in the 4 corners game, denote this (1,2,3,4) ). This is the identity position in D8. But this can also be considered the identity in a specific some group of S4 (which we will calculate). Rotating once, we get (4,1,2,3). So 1 went to 2, 2 went to 3, 3 went to 4, and 4 went to 1. So this is the element (1 2 3 4) of S4. Rotating again, we get (3, 4, 1, 2). So now we have (1 3)(2 4) of S4.

So r maps to (1 2 3 4) and r^2 maps to (1 3)(2 4). Work about the entire correspondence for D4 and prove it is a bijection. Moreover, prove that all the elements of S4 that the D4 elements map to form a subgroup.

4. Prove that any finite sequence of english letters (e.g ABCDDDME) has a bijection with the natural numbers that are greater than or equal to 2 (hint, think the RSA presentation with Wesley, James, and Prannay. They used power of primes multiplied together with fundamental theorem of arithmetic to do this).

5. Prove that there is a bijection between the natural numbers (include 0) and the infinite set (10, 11, 12, 13, 14, 15,...)

6. Consider a cubic polynomial, ax^3 + bx^2 +cx + d. Show that there is a bijection from the set of all cubic polynomials to the set of all 4 tuples of reals (e.g (0,0,2,4)).

Ch 7: 7.1, 7.3, 7.5, 7.8

Homework 4:

Ch 13: 13.1, 13.2, 13.3, 13.6 (Q here is the quaternions and not the rationals. James proved there are non cyclic subgroups of the rationals : ) ).

Ch 11: 11.12 (Now, try to use Cauchy's theorem. If you cannot get it, email Julian for a hint).