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Is every abelian group is cyclic or not and why?
every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic.
Every subgroup of a cyclic group is cyclic?
Yes, every subgroup of a cyclic group is cyclic because every subgroup is a group.
Is every solvable group abelian?
No.
Is the symmetry group of the square an abelian group?
Abelian meaning commutative. If the symmetry group of a square is commutative then it's an abelian group or else it's not.
Proof or Disprove 'If every proper subgroup of G is cyclic then G must be cyclic'?
No! Take the quaternion group Q_8.
Is every abelian group is cyclic or not and why?
every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic.
Is every finite abelian group is cyclic?
No, for instance the Klein group is finite and abelian but not cyclic. Even more groups can be found having this chariacteristic for instance Z9 x Z9 is abelian but not cyclic
Is every cyclic group abelian?
Yes. Lets call the generator of the group z, then every element of the group can be written as zk for some k. Then the product of two elements is: zkzm=zk+m Notice though that then zmzk=zm+k=zk+m=zkzm, so the group is indeed abelian.
Every subgroup of a cyclic group is cyclic?
Yes, every subgroup of a cyclic group is cyclic because every subgroup is a group.
Is every solvable group abelian?
No.
Prove that a group of order three is abelian?
By LaGrange's Thm., the order of an element of a group must divide the order of the group. Since 3 is prime, up to isomorphism, the only group of order three is {1,x,x^2} where x^3=1. Note that this is a finite cyclic group. Since all cyclic groups are abelian, because they can be modeled by addition mod an integer, the group of order 3 is abelian.
Is the symmetry group of the square an abelian group?
Abelian meaning commutative. If the symmetry group of a square is commutative then it's an abelian group or else it's not.
What is the definition of an abelian group?
An abelian group is a group in which ab = ba for all members a and b of the group.
Prove that every a cyclic group is an abelia?
Let G be the cyclic group generated by x, say. Ten every elt of G is of the form x^a, for some a
What is an Abelian?
An abelianization is a homomorphism which transforms a group into an abelian group.
Is a group of order 24 abelian group or not?
The abelian groups of order 24 are C3xC8, C2xC12, C2xC2xC6. There are other 12 non-abelian groups of order 24
Proof or Disprove 'If every proper subgroup of G is cyclic then G must be cyclic'?
No! Take the quaternion group Q_8.
