Test by Bikram | Mock GATE | Test 2 | Question: 21

Question

Which of the following statements are false?

1. The order of any finite group is always divisible by the order of the subgroups.
2. Intersection of two subgroups of a group $G$, may or may not be a subgroup of $G$.
3. Proper subgroup of an infinite cyclic group is infinite.
4. Prime order group has both proper and improper subgroups.

• A. III and IV
• B. II and IV
• C. I and II
• D. II, III, IV

Comments

@shiva and @Niharika 1

please read the best chosen answer.

also  to know why B is incorrect 

http://math.stackexchange.com/questions/702186/prove-that-the-intersection-of-two-subgroups-is-a-subgroup

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sir can u plz give any argument or ref for stmnt3???

All proper subgrp may not infinte.

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For Statement 3 to be true, it should have been "Every non trivial subgroup of infinite cyclic group is infinite". As for the given statement in question,{e} is a finite proper subgroup. So it is false.

Answer 1

 statement 1:--  The order of a group is always divisible by the order of the subgroups. its is always coreect
stmt 2;-  Intersection of two subgroups of a group G, may or may not be a subgroup of G. its incorrect it is always subgroup but union may or may not

stmt 3---  Proper subgroup of an infinite cyclic group is infinite. yes true

 stmt 4;--Prime order group has both proper and improper subgroups. false as prime order group has no proper subgroup 

https://proofwiki.org/wiki/Prime_Group_has_no_Proper_Subgroups so stament II and IV is wrong hence (B) is correct 

Comments

example for why union of two subgroups is not a subgroup -

two subgroups { -1 , 0 , 1 } and {-2, 0 , 2} are closed under addtion.

Union  = { -2 , -1 , 0 , 1 , 2 } is not closed as 1+2 = 3 which is not in the group.

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Prime order group has No Proper Subgroups except the trivial subgroup ({e},*).

https://proofwiki.org/wiki/Prime_Group_has_no_Proper_Subgroups

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1st statement is true only for finite sets. That's Lagrange's theorem

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how can you say proper subgroup of infinite cyclic group is infinite? Isn’t {e} proper subgroup of some infinite cyclic group? {e} trivial as well as proper subgroup.of any group. If the statement was “ Every non trivial subgroup of infinite cyclic group is infinite” then it should be true.