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MATH 150A: HOMEWORK #2 Key concepts: composition law, definition of a group

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MATH 150A: HOMEWORK #2

Key concepts: composition law, definition of a group. examples and non-examples of
groups, subgroups, examples of subgroups. subgroups of the integers, Euclidean algo-
rithm, cyclic subgroups, order of an clement

e Read sections 2.1-2.4 from Artin.

* Written assignment: Do exercises 2.6 (20 points), 3.1 (20 points) and 4.3 (20
points) on pages 69-70 of Artin and the exercise below (40 points).

A. For each of the following, decide wether it is a group or not a group. If it is, briefly
explain why. If not, just state which axiom it fails.

(1) The set of strictly positive real numbers with multiplication.

(2) The set of injective functions f : {1,2,....2} -> {1,2.....2}, with the operation
being composition of functions.

(3) The set of injective functions f : Z -+ Z, again with composition of functions.

(4) The set of even permutations of {1,2,....n}, typically denoted A,, with product
of permutations.

(5) The set {-10,-9,-8,...,8,9,10} with (usual) addition.

(6) The set {black, white} with the composition law * defined by whitexwhite = white,
white « black = black, black « white = black, black » black = black.

(7) The set {black. white} with the composition law * defined by whitexwhite = white,
white * black = black, black * white = black, black * black = white.

(8) The set of so-called orthogonal matrices, i.c.

{A: Aisa real n xn matrix such that A- A‘ = I}

with matrix multiplication. Recall that J, denotes the identity matrix and A! is the
transpose matrix of A.
 

2.6. The matrix below is based on the Pascal triangle. Find its inverse.
1
1 1
12 1 ,
1 3 3 1
146 4 1

3.1. A matrix B is symmetric if B = B'. Prove that for any square matrices B, BB‘ and B + B'
are symmetric, and that if A is invertible, then (A~!)' = (A')7.

4.3. Compute the determinant of the following n Xn matrix using induction on n:
2 -1
-1 2 -]
-1 2-1
-l - ,
2 -1
-1 2