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 , use the group Z 2 with g = 1 and h = 1 .
For n = 2 , use the group Z 2 × Z 2 with g = ( 1 , 0 ) and h = ( 0 , 1 ) .
For 1"> n > 1 , use the dihedral group D 2 n with g = s and h = rs .
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 .
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 = 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 = ( 1 , 0 ) + ( 0 , 1 ) = ( 1 , 1 ) , so ∣ g h ∣ = 2 .
Case n>1 For 1"> n > 1 , consider the dihedral group D 2 n . The dihedral group D 2 n can be presented as ⟨ r , s ∣ r n = s 2 = 1 , srs = r − 1 ⟩ . Let g = s and h = rs . Then ∣ g ∣ = 2 and ∣ h ∣ = 2 . We have g h = s ( rs ) = srs = r − 1 . Thus, ∣ g h ∣ = ∣ r − 1 ∣ = ∣ r ∣ = n .
Conclusion Therefore, for n = 1 , we can use Z 2 with g = 1 and h = 1 . For n = 2 , we can use Z 2 × Z 2 with g = ( 1 , 0 ) and h = ( 0 , 1 ) . For 1"> n > 1 , we can use D 2 n with g = s and h = rs .
Examples
Understanding group theory helps in various fields like cryptography, coding theory, and physics. For instance, in cryptography, elliptic curve cryptography relies on the properties of elliptic curves defined over finite groups. The structure of these groups determines the security of the encryption scheme. Similarly, in physics, group theory is used to classify elementary particles and understand their interactions.
For each positive integer n , we can construct groups with elements of order 2 such that the product's order equals n . Specifically, for n = 1 use Z 2 , for n = 2 use Z 2 × Z 2 , and for 1"> n > 1 use the dihedral group D 2 n . This approach clearly demonstrates how different group structures accommodate the requirements of the problem.
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