Let G be a finite group. Show that if G has exactly one nontrivial subgroup, then order of G is
p^2 for some prime p.
Solution
Lagrange \'s theorem in group theory states that :
( G ,* ) is a group , His a sub group of G then O (H) divides O(G)
using the above fact we conclude that :
If O(G) = n then order of the subgroups are the factors of \'n\'
ie if 1, a, b , ------n are the distinct factors of \'n
then there are subgroups of order 1,a,b, ---n
the proper subgroups will have order a,b ,
there is only one subgroup => the factors of \'n\' are 1,a,n => 1xn=a2 =p2 where \'p \' is a prime
no
Example n = 9 only one proper subgroup of order =3
n= 25 ---------------- \'\'-----------------------=5 and so on.
matching questiona)africa b)Asia c)America d)lesser ape .pdf
1. matching question:
a)africa b)Asia c)America d)lesser ape e)great ape
Geographic location category
1)Hylobate
2)Pongo
3)Gorilla
a)flowers b)fruit c) leaves d)everything e)grass
1)Folivorous primates eat
2)Omnivorous primates eat
3)Frugivorous primates east
Solution
Matching question:
1) Hylobate- geographical location is b) Asia.
- belong to category d) lesser ape.
2) Pongo- geographical location is b) Asia. They are native to Indonesia.
- belong to category e) great ape.
3) Gorilla- geographical location is a) Africa. They inhabit parts of central Africa.
- belong to category e) great ape.
1) Folivorous primates eat c) leaves.
2) Omnivorous primates eat d) everything.
3) Frugivorous primates eat b) fruit.
