Let G be the cyclic group generated by x, say. Ten every elt of G is of the form x^a, for some a
There's a theorem to the effect that every group of prime order is cyclic. Since 5 is prime, the assertion in the question follows from the said theorem.
Yes. The only group of order 1 is the trivial group containing only the identity element. All groups of orders 2 or 3 are cyclic since 2 and 3 are both prime numbers. Therefore, any group of order less than or equal to four must be a cyclic 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.
Four of them.
D2, the dihedral group of order 4, consists of rotations and reflections of a square, while Z12, the cyclic group of order 12, is generated by the addition of integers modulo 12. D2 is not cyclic, as it cannot be generated by a single element, whereas Z12 is cyclic, generated by 1. Furthermore, the structure of the groups differs: D2 has elements of order 2 (the reflections) and elements of order 4 (the rotations), while Z12 has elements of various orders that are consistent with a cyclic structure. Hence, their different algebraic structures confirm that D2 and Z12 are not isomorphic.
Yes, every subgroup of a cyclic group is cyclic because every subgroup is a group.
every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic.
There's a theorem to the effect that every group of prime order is cyclic. Since 5 is prime, the assertion in the question follows from the said theorem.
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
No! Take the quaternion group Q_8.
A cyclic group of order two looks like this.It has two elements e and x such that ex = xe = x and e2 = x2 = e.So it is clear how it relates to the identity.In a cyclic group of order 2, every element is its own inverse.
Yes. The only group of order 1 is the trivial group containing only the identity element. All groups of orders 2 or 3 are cyclic since 2 and 3 are both prime numbers. Therefore, any group of order less than or equal to four must be a cyclic group.
A cyclic group, by definition, has only one generator. An example of an infinite cyclic group is the integers with addition. This group is generated by 1.
No Q is not cyclic under addition.
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.
Cyclic photophosphorylation is when the electron from the chlorophyll went through the electron transport chain and return back to the chlorophyll. Noncyclic photophosphorylation is when the electron from the chlorophyll doesn't return back but incorporated into NADPH.