Sage-Tips

Is Z4 a cyclic group?

Both groups have 4 elements, but Z4 is cyclic of order 4. In Z2 × Z2, all the elements have order 2, so no element generates the group.

Is Z3 Z3 cyclic?

(d) • Z3 is cyclic, generated additively by 1: We have [1], and [1] + [1] = [1+1] = [2] and [1]+[1]+[1] = [1+1+1] = [0] = e, so all elements are captured. Z2 × Z2 is not cyclic: There is no generator.

Is Z4 Z15 cyclic group?

Any non-identity element in Z2 ⊕ Z2 has order 2. There are only 3 elements of order 2 in Z4 ⊕ Z4: (2,0), (0,2), and (2,2). Therefore there is no cyclic subgroup of order 9 in Z12 ⊕ Z4 ⊕ Z15.

What is cyclic group in discrete mathematics?

A cyclic group is a group that can be generated by a single element. Every element of a cyclic group is a power of some specific element which is called a generator. A cyclic group can be generated by a generator ‘g’, such that every other element of the group can be written as a power of the generator ‘g’.

Is Q cyclic group?

The additive group of rational numbers (Q, +) is locally cyclic – any pair of rational numbers a/b and c/d is contained in the cyclic subgroup generated by 1/(bd).

Is the group Z4 Z6 cyclic?

Yes, this group is cyclic. For example, (1,1) is a generator. 7. The order of 3 in Z4 is 4; the order of 6 in Z12 is 2; the order of 12 in Z20 is 5; the order of 16 in Z24 is 3.

What does it mean for a group to be cyclic?

A cyclic group is a group that can be generated by a single element. (the group generator). Cyclic groups are Abelian.

What is Z2 group?

It is the only finite group with exactly two conjugacy classes. There are, however, infinite groups with exactly two conjugacy classes; see group with two conjugacy classes. The group is a group of prime order, hence its only subgroups are itself and the trivial subgroup.

Is Z3 Z4 cyclic?

From this, you can see that the group Z3 × Z4 is cyclic because it can be generated by a single element. so from this you can see that Z2 × Z12 ∼ = Z4 × Z6. (24) List all finite abelian groups of order 720, up to isomorphism.

Is Z3 * Z4 cyclic?

Note. So we see that Z3 × Z4 is a cyclic group of order 12.