State true or false with brief justification b Every free gr

State true or false with brief justification:-

b) Every free group has infinitely many generators

Solution

a) Yes O_2,n-2 is a subgroup of SLn(R). Because SLn(R) is a lie subgroup of special linear subgroup which happesn to contain all nXn matrices over R with vanishing traces.

b) False. Not all free groups have infinitely many generators.

c) False. R is a semigroup under addition, therefore, this statement is false.