For each of the following sets, determine whether they form a group under the op- eration...

  

Transcribed Image Text:

For each of the following sets, determine whether they form a group under the op- eration indicated (where it is relevant you may assume that matrix multiplication is associative): (a) the integers (mod 10) under addition; (b) the integers (mod 10) under multiplication; (c) the integers 1,2,3,4,5,6 under multiplication (mod 7); (d) the integers 1.2.3.4.5 under multiplication (mod 6); (e) all matrices of the form a ( a - b). 0 where a and b are integers (mod 5) and a 0+b, under matrix multiplica- tion: (f) those elements of the set in (e) that are of order 1 or 2 (taken together); (g) all matrices of the form 100 a 10 b C 1 where a, b, c are integers, under matrix multiplication. For each of the following sets, determine whether they form a group under the op- eration indicated (where it is relevant you may assume that matrix multiplication is associative): (a) the integers (mod 10) under addition; (b) the integers (mod 10) under multiplication; (c) the integers 1,2,3,4,5,6 under multiplication (mod 7); (d) the integers 1.2.3.4.5 under multiplication (mod 6); (e) all matrices of the form a ( a - b). 0 where a and b are integers (mod 5) and a 0+b, under matrix multiplica- tion: (f) those elements of the set in (e) that are of order 1 or 2 (taken together); (g) all matrices of the form 100 a 10 b C 1 where a, b, c are integers, under matrix multiplication.