60 votes 60 votes How many different non-isomorphic Abelian groups of order $4$ are there? $2$ $3$ $4$ $5$ Set Theory & Algebra gatecse-2007 group-theory normal + – Kathleen asked Sep 21, 2014 Kathleen 19.6k views answer comment Share Follow See all 13 Comments See all 13 13 Comments reply Show 10 previous comments ankitgupta.1729 commented Dec 4, 2019 i edited by ankitgupta.1729 Dec 7, 2019 reply Follow Share I could not resist myself to comment again here because this question is quite related to the great indian mathematician Srinivasa Ramanujan. S. Ramanujan worked on partitions with G.H. Hardy and obtained the asymptotic solution for partition function : $P(n) \sim \frac{1}{4n\sqrt{3}}\; e^{\pi \sqrt{2n/3}}$ I have uploaded some video clips from the movie "The Man who knew infinity". Anyone who is interested in mathematics can see these video clips. 1) Partitions 2) Ramanujan with P.C. Mahalonobis(Founder of ISI, father of indian statistics and a FRS too like Ramanujan) 3) Extra 4) Number 1729 17 votes 17 votes Verma Ashish commented Dec 19, 2019 reply Follow Share I watched that movie 2years back. But didn't noticed a single point about partitions.. Thanks for sharing.. :) 6 votes 6 votes vizzard110 commented Jan 11, 2020 reply Follow Share That time I didn't even knew the importance of partitions. 1 votes 1 votes Please log in or register to add a comment.
–2 votes –2 votes Option b . anshu answered Jan 30, 2015 anshu comment Share Follow See all 5 Comments See all 5 5 Comments reply ankitrokdeonsns commented Apr 24, 2015 reply Follow Share can you explain pls 0 votes 0 votes Digvijay Pandey commented Apr 25, 2015 reply Follow Share 1. Group with order <=5 always abelian. if we find no of groups of order 4 that ll be same as no of abelian group of order 4.. in every group of order d , so element order 1 and 4 common in every group.. d only thing dat matters is no of groups with or without order 2. 2 non isomorphic group of order 4.. 3 votes 3 votes Kaluti commented Jul 23, 2017 reply Follow Share First, find the prime factorization of n. For example, 4 has prime factorization as 2*2. Also, 600 can be factorized as 2^(3)∗3^(1)∗5^(2) Now, find the number of partitions of all powers, and then multiply them. Number of Partitions of a number k is the number of ways k can be partitioned. For example, number of partitions of 3 is 3, because 3 can can be partitioned in 3 different ways : {1+1+1}, {1+2}, {3}. Similarly, 4 can be partitioned in 5 different ways : {1+1+1+1},{2+1+1},{2+2},{3+1},{4}. Note that order of elements in a partition doesn't matter, so for example, partitions {2+1+1} {1+1+2} are same. So for this question, we will find number of partitions of 2, which is 2 : {1+1},{2}. There is no other power, so answer is 2 only. Suppose, in question, order given is 600, , so different powers are 3,1,2. Number of partitions for 3,1,2 are 3,1,2 respectively, so result would have been 3∗1∗2=6. 16 votes 16 votes reena_kandari commented Sep 6, 2017 reply Follow Share what is meant by isomorphic and non-isomorphic abelian group,did't get any good reference. @Kaluti any help? 1 votes 1 votes Kaluti commented Sep 7, 2017 i edited by Puja Mishra Jan 20, 2018 reply Follow Share group theorists view two isomorphic groups as follows: For every element g of a group G, there exists an element h of H group such that h 'behaves in the same way' as g(operates with other elements of the group in the same way as g). For instance, if g generates G, then so does h. This implies in particular that G and H are in bijective correspondence. Thus, the definition of an isomorphism is quite natural. you can refer this 7 votes 7 votes Please log in or register to add a comment.