Mathematics with Maple

S.Duzhin

Solutions for Lesson 10.

Exercise 1. Check that the set B that consists of the six different roots of 1 of degree 6 (i.e. all complex numbers x such that x^6=1) makes a group.

> B:=[solve(x^6=1,x)];

> for i from 1 to 6 do

> for j from 1 to 6 do print(expand(B[i]*B[j]));

> od;

> od;

Exercise 2.

How many different functions can you obtain from

f(x) = 1/x and g(x) = (x-1)/(x+1) using the composition of functions?

Answer: eight, their list is

x, 1/x, -x, -1/x, (x-1)/(x+1), (x+1)/(x-1), -(x-1)/(x+1), -(x+1)/(x-1).

This list can be obtained using the Maple commands like

> f:=x->1/x;

> g:=x->(x-1)/(x+1);

> simplify(g(g(x)));

> simplify(g(g(g(x))));

> simplify(g(g(g(g(x)))));

> simplify(f(g(x)));

> simplify(f(g(g(x))));

> simplify(g(f(x)));

Exercise 3. Get help about mulperms (?mulperms;) and compute the product of the permutations [1,3,2] and [1,3] using it. Then make the same computation by hand and compare.

Answer: [2,3] (transposition).

Solution:

> with(group):

> mulperms([[1,3,2]],[[1,3]]);

Exercise 4. Define the group of all permutations of degree 4, find its order and th list of elements.

Answer: the order is 24.

Solution:

> S4:=permgroup(4, {[[1,2]], [[2,3]], [[3,4]]});

> U:= permgroup(4, {[ ]});

> cosets(S4,U);

Exercise 5. How many different permutations of degree 7 can be obtained taking the products of the two permutations [1,2,3,4] and [4,5,6,7]?

Answer: 5040 (i.e. all permutations on 7 symbols).

Solution:

> G:=permgroup(7, {[[1,2,3,4]], [[4,5,6,7]]});

> grouporder(G);

Exercise 6. Define the group with two generators, s and v, and three relations:

s^3=e, v^7=e, s*v*1/s = v^2. Find the number of elements in this group.

Answer: the group consists of 21 elements.

Solution:

> SV:=grelgroup({s,v}, {[s,s,s],[v,v,v,v,v,v,v],[s,v,1/s,1/v,1/v]});

> grouporder(SV);