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
The abelian groups of order 24 are C3xC8, C2xC12, C2xC2xC6. There are other 12 non-abelian groups of order 24
The set of integers, under addition.
Normally, a cyclic group is defined as a set of numbers generated by repeated use of an operator on a single element which is called the generator and is denoted by g.If the operation is multiplicative then the elements are g0, g1, g2, ...Such a group may be finite or infinite. If for some integer k, gk = g0 then the cyclic group is finite, of order k. If there is no such k, then it is infinite - and is isomorphic to Z(integers) with the operation being addition.
rectangle has inversion (180 deg rotation) hexagon has 60 deg ratation, cyclic group genterated is 60, 120, 180, 240, 300, 360=0 equilateral triangle has 120 deg rotation, cyclic group genterated is 120, 240, 360=0
every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic.
No.
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
Yes, every subgroup of a cyclic group is cyclic because every subgroup is a group.
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.
No.
Abelian meaning commutative. If the symmetry group of a square is commutative then it's an abelian group or else it's not.
An abelian group is a group in which ab = ba for all members a and b of the group.
Let G be the cyclic group generated by x, say. Ten every elt of G is of the form x^a, for some a
An abelianization is a homomorphism which transforms a group into an abelian group.
The abelian groups of order 24 are C3xC8, C2xC12, C2xC2xC6. There are other 12 non-abelian groups of order 24
No! Take the quaternion group Q_8.