29 votes 29 votes Let $G$ be a finite group on $84$ elements. The size of a largest possible proper subgroup of $G$ is _____ Set Theory & Algebra gatecse-2018 group-theory numerical-answers set-theory&algebra 1-mark + – gatecse asked Feb 14, 2018 • retagged Dec 1, 2022 by Lakshman Bhaiya gatecse 12.3k views answer comment Share Follow See all 2 Comments See all 2 2 Comments reply akash.dinkar12 commented Feb 14, 2018 reply Follow Share 42..... 0 votes 0 votes Mk Utkarsh commented Feb 25, 2018 reply Follow Share https://www.youtube.com/watch?v=TCcSZEL_3CQ 3 votes 3 votes Please log in or register to add a comment.
Best answer 45 votes 45 votes Order of a Subgroup always divides the order of Group. Proper Subgroup of Group having order $84$ would have one of the order (proper factors of $84)$ $2,3, 4,6,7,12,14, 21,28, 42$. So the largest order would be $42$. Digvijay Pandey answered Feb 14, 2018 • edited Jun 14, 2021 by Arjun Digvijay Pandey comment Share Follow See all 10 Comments See all 10 10 Comments reply Ashish01 commented Jul 14, 2018 reply Follow Share group also itself called subgroup then why 84 not true? 1 votes 1 votes Neeraj_gate19 commented Jul 15, 2018 reply Follow Share group also itself called subgroup but that considered as trivial subgroup. Proper subgroup is a subgroup which is not trivial . Also a group consist of only identity element is considered as trivial subgroup. 7 votes 7 votes akash.dinkar12 commented Aug 7, 2018 i edited by akash.dinkar12 Aug 7, 2018 reply Follow Share Divisors of the Positive Integer 84. 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 All are possible proper subgroups excluding the size of 1 and 84 because they are trivial subgroups but here the question is asking, Largest possible proper subgroup possible is 42... 21 votes 21 votes Lakshman Bhaiya commented Jan 10, 2019 reply Follow Share @akash.dinkar12 The size of the smallest proper subgroup of $G=2?$ 1 votes 1 votes akash.dinkar12 commented Jan 11, 2019 reply Follow Share Yes it is the smallest non trivial group 1 votes 1 votes learncp commented Nov 10, 2019 reply Follow Share Will the number of subgroups possible be $\Phi (84)$ = 24 ? 0 votes 0 votes Lakshman Bhaiya commented Nov 10, 2019 reply Follow Share $\Phi(84) = 24$, number of generator,it is valid on cyclic group. see this https://math.stackexchange.com/questions/205918/determining-number-of-subgroups 0 votes 0 votes nishant_magarde commented Jul 15, 2020 reply Follow Share So here trivial subgroups can be identity elements and the group itself. Right? 0 votes 0 votes Lakshman Bhaiya commented Jul 16, 2020 reply Follow Share Yes 0 votes 0 votes Akash 15 commented Dec 27, 2023 reply Follow Share Let $G$ be a finite group on $84$ elements. The size of a largest possible $\text{proper}$ subgroup of $G$ is $\color{Red}42$ Let $G$ be a finite group on $84$ elements. The size of a largest possible subgroup of $G$ is $\color{Red}84$ (Group itself) 0 votes 0 votes Please log in or register to add a comment.
17 votes 17 votes Lagrange's theorem states that order of every subgroup of G, it must be the divisor of G. So the largest subgroup will be 84 which is trivial, but in the question it is asking for the proper subgroup hence it will be 42. Reference: https://en.wikipedia.org/wiki/Subgroup Prashant Kumar 4 answered Feb 4, 2018 Prashant Kumar 4 comment Share Follow See all 2 Comments See all 2 2 Comments reply raviyogi commented Feb 5, 2018 reply Follow Share got same 0 votes 0 votes Kaluti commented Mar 4, 2018 reply Follow Share but for prime divisor we are sure that it would be proper subgroup for other divisor it may or not be proper subgroup converse of lagrange's theorem is not true 1 votes 1 votes Please log in or register to add a comment.
4 votes 4 votes Order of Group must be divisible by order of subgroup = 42. Prashant. answered Feb 14, 2018 Prashant. comment Share Follow See all 0 reply Please log in or register to add a comment.
1 votes 1 votes 42 is correct RFITNES. TK answered Feb 14, 2018 RFITNES. TK comment Share Follow See all 0 reply Please log in or register to add a comment.