For every positive integer $n$, construct a group containing elements $g, h$ such that $|g|=2,|h|=2,|g h|=n$. (Hint: For $n>1, D_{2 n}$ will do.)
Answer (2)
For n = 1 , the group Z 2 with g = 1 and h = 1 satisfies ∣ g ∣ = 2 , ∣ h ∣ = 2 , and ∣ g h ∣ = 1 .
For n = 2 , the group Z 2 × Z 2 with g = ( 1 , 0 ) and h = ( 0 , 1 ) satisfies ∣ g ∣ = 2 , ∣ h ∣ = 2 , and ∣ g h ∣ = 2 .
For 2"> n > 2 , the dihedral group D 2 n with g = s and h = rs satisfies ∣ g ∣ = 2 , ∣ h ∣ = 2 , and ∣ g h ∣ = n .
Thus, for every positive integer n , we can construct a group containing elements g and h such that ∣ g ∣ = 2 , ∣ h ∣ = 2 , and ∣ g h ∣ = n .
Explanation
Understanding the Problem We are asked to construct a group for every positive integer n . The group must contain elements g and h such that the order of g is 2, the order of h is 2, and the order of g h is n . The hint suggests using the dihedral group D 2 n for 1"> n > 1 . The order of an element x is denoted by ∣ x ∣ . The dihedral group D 2 n is the group of symmetries of a regular n -gon, and it has 2 n elements.
Case n=1 For n = 1 , consider the group Z 2 = { 0 , 1 } under addition modulo 2. Let g = 1 and h = 1 . Then ∣ g ∣ = 2 , ∣ h ∣ = 2 , and g h = g + h = 1 + 1 = 0 , so ∣ g h ∣ = 1 .
Case n=2 For n = 2 , consider the group Z 2 × Z 2 = {( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 )} under component-wise addition modulo 2. Let g = ( 1 , 0 ) and h = ( 0 , 1 ) . Then ∣ g ∣ = 2 , ∣ h ∣ = 2 , and g h = g + h = ( 1 , 1 ) , so ∣ g h ∣ = 2 .
Case n>2 For 2"> n > 2 , consider the dihedral group D 2 n . The elements of D 2 n can be written as r i s j where 0 ≤ i < n and 0 ≤ j < 2 , where r is a rotation by 2 π / n and s is a reflection. We have the relations r n = s 2 = 1 and srs = r − 1 . Let g = s and h = rs . Then ∣ g ∣ = 2 and ∣ h ∣ = 2 since ( rs ) ( rs ) = r ( srs ) s = r ( r − 1 ) s 2 = e . Then g h = s ( rs ) = srs = r − 1 . Thus ∣ g h ∣ = ∣ r − 1 ∣ = ∣ r ∣ = n . Therefore, for 2"> n > 2 , the dihedral group D 2 n with g = s and h = rs satisfies the conditions.
Examples
This problem demonstrates how group theory, an abstract branch of mathematics, can be used to describe symmetries and transformations. Understanding group structure is crucial in various fields such as physics (describing symmetries of particles), chemistry (analyzing molecular structures), and computer science (designing efficient algorithms). The dihedral group, in particular, is used to model the symmetries of regular polygons, which has applications in computer graphics and robotics.
We can construct a group for every positive integer n containing elements g and h where ∣ g ∣ = 2 , ∣ h ∣ = 2 , and ∣ g h ∣ = n . For n = 1 , we use ma t hbb Z 2 ; for n = 2 , ma t hbb Z 2 × Z 2 ; and for 2"> n > 2 , the dihedral group D 2 n . Each of these groups satisfies the required properties for the elements defined.
;
