Find two elements of maximum order in the group G = Z100 e Z4O Z2.How many such elements are there?The number of abelian groups of order 2ª36 is equal to the number ofabelian groups of order 2°3". Can you generalize this observation?Let G = Z4 O Z4 O Z25 O Z5 O Z7. Find the corresponding "invariantfactor form".

As Z100 has no element of order 100 otherwise it will be cyclic which is not true. So, we consider the element 2 in Z100 which has the highest order in the group. Order (2) in Z100 is 50. Similarly, highest order of an element in Z4 is 2 of the elements 1, 3. .Highest order of an element in Z2 is 2 that is of the element 1. Finally, to compute the maximum order of an element in the group G we have to take the lcm of the orders of the elements in the group Z100 , Z4 and Z2 that is lcm ( |2|, |1|, |1| ) = lcm ( 50, 2, 2 ) = 50. Hence, the highest order of the element in the group G is 50 and elements are ( 2, 1, 1 ) and ( 2, 3, 1 ) separated by the direct sum operation. From the fundamental theorem of Abelian groups we know for a fixed integer n every Abelian group of order n can be written into direct sum of cyclic groups in s finite way upto isomorphism. Now, we can count the number of such groups.. If prime decomposition of n is given as n = ( p1 ^ a1 ) × ( p2 ^ a2 ) × ... × ( pn ^ an ) , where p1, p2, ..., pn are primes and a1, a2, ..., an are non - negative integers. Then, total number of Abelian groups of order n are p( a1 ) × p( a2 ) × ... × p( an ), where p denotes the number of partitions of n. Now, number of Abelian groups of order (2^a)(3^b) and (2^b)(3^a) are same because they are both equal to p(a) × p(b) .Note that ds denote the direct sum. Now, Z 4 ds Z 4 ds Z 25 ds Z 5 ds Z 7 is isomorphic to Z 100 ds Z 140 using the fact Z mn is isomorphic to Z m ds Z n if gcd (m,n) = 1. Hence, invariant factor form is Z 100 ds Z 140 as Z 4 ds Z 25 = Z 100 and Z 4 ds Z 5 ds Z 7 = Z 20 ds Z 7 = Z 140 and gcd( 100, 140 ) = 20 so we can't proceed further. Therefore, we obtained our desired results.

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Find two elements of maximum order in the group G = Z100 Z4 O Z2. How many such elements are there? The number of abelian groups of order 2"36 is equal to the number of abelian groups of order 2°3ª. Can you generalize this observation? Let G = Z4 O Z4 O Z25 O Z5 O Z7. Find the corresponding “invariant factor form".

Find two elements of maximum order in the group G = Z100 Z4 O Z2. How many such elements are there? The number of abelian groups of order 2"36 is equal to the number of abelian groups of order 2°3ª. Can you generalize this observation? Let G = Z4 O Z4 O Z25 O Z5 O Z7. Find the corresponding “invariant factor form".

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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