THE GROUPS WHICH ARE GENERATED BY TWO 



OPERATORS OF ORDERS TWO AND FOUR 



RESPECTIVELY WHOSE COMMUTATOR 



IS OF ORDER TWO. 



By G. a. MILLER. 



(Read April ig, 1907.) 



We shall first give two general theorems relating to commu- 

 tators^ which will be used in what follows. 



Theorem I. If a commutator is commutative zvith one of its 

 elements the order of the commutator divides the order of this 

 element. 



Theorem 11. The smallest invariant subgroup ivhich contains the 

 commutator of tzvo operators includes the commutator subgroup of 

 the group generated by these operators. 



To prove the former of these two theorems, it is only necessary 

 to observe that if c^=s^~^s^''^s^s^ is commutative with s^ or So it is 

 also commutative with ^2~^-^i^2 oi" ^{'^^2^1^ respectively. Hence ^1 or s^ 

 would be commutative with sf'^s^So or s^-^^s^s^, respectively. The proof 

 of the theorem follows now^ from the facts that the order of the 

 product of two commutative operators divides the least common 

 multiple of their orders, and that the orders of the two factors in 

 the present case are the same. The proof of the second theorem 

 follows from the fact that the two operators in question would cor- 

 respond to commutative operators in the quotient group with respect 

 to the given invariant subgroup. In particular we have the corol- 

 lary, // two operators generate a group the conjugates of their com- 

 mutator generate the commutator subgroup. 



Let .^1, ^2 be any two operators of orders 2 and 4, respectively, 

 which satisfy the conditions 



s^^=i, s.^=i, s^s.-:s^s^ = s^^s-^s.^s^, 



^ The principal known theorems relating to commutators are found in two 

 articles published in the Bulletin of the American Mathematical Society, vol. 

 4, p. 135 and vol. 6, p. 105. 



146 



