What are all of the cyclic subgroups of the quaternion group?

Tables classifying subgroups up to automorphisms

What are all of the cyclic subgroups of the quaternion group Q8?

1. The subgroups of Q8 are: {1} {1, −1} {1, i, −1, −i} {1, j, −1, −j} {1, k, −1, −k} Q8 The commutator subgroup contains the element [i, j] = iji−1j−1 = ij(−i)(−j)=(ij)(ij) = k2 = −1. Similarly [j, k] = −1 and [k, i] = −1.

How do you find the subgroups of quaternion groups?

Quaternion group Q8 = {1 , -1 , i, -i, j, -j, k, -k} Trivial subgroups – Q8 , {1} . proper subgroups – Z(Q8) ={1, -1} , = { 1, -1, i, -i} , = {1, -1, j, -j} , ={1, -1, k, -k}.) Let G be a finite group and H subgroup of G, which is normal subgroup if and only if index of H in G is 2.

How many cyclic subgroups are in Q8?

six subgroups
Thus the six subgroups of Q8 are the trivial subgroup, the cyclic subgroups generated by −1, i, j, or k, and Q8 itself. (2b) Find Z(Q8). SOLUTION: The identity element is always contained in the center of a group, so we wish to know if any other elements of Q8 lie in the center.

What is cyclic group in group theory?

In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group that is generated by a single element. This element g is called a generator of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers.

Is quaternion group cyclic?

The generalized quaternion groups have the property that every abelian subgroup is cyclic.

Are the quaternions cyclic?

Thus, representation of quaternion group Q contains cyclic normal subgroups N3, N4, and N5 such that factor groups Q/N3, Q/N4, and Q/N5 are also cyclic.

Is Z8 cyclic?

Z8 is cyclic of order 8, Z4 ×Z2 has an element of order 4 but is not cyclic, and Z2 ×Z2 ×Z2 has only elements of order 2.

Is the quaternion group cyclic?

What is the cyclic group of order 2?

The cyclic group of order 2 may occur as a normal subgroup in some groups. Examples are the general linear group or special linear group over a field whose characteristic is not 2. This is the group comprising the identity and negative identity matrix. It is also true that a normal subgroup of order two is central.

How do you find the subgroup of a cyclic group?

Theorem (1): If G is a finite cyclic group of order n and m∈N, then G has a subgroup of order m if and only if m|n. Moreover for each divisor m of n, there is exactly one subgroup of order m in G.

Which of the following is a cyclic group?

A cyclic group is a group which is equal to one of its cyclic subgroups: G = ⟨g⟩ for some element g, called a generator. For a finite cyclic group G of order n we have G = {e, g, g2, , gn−1}, where e is the identity element and gi = gj whenever i ≡ j (mod n); in particular gn = g0 = e, and g−1 = gn−1.

What is the Order of the quaternions of a subgroup?

Looks correct. By Lagrange’s Theorem, a consequence of Cauchy’s Theorem the order of a subgroup must divide the order of the group. Since the order of the quaternions is 8 this means any proper, nontrivial subgroup must be of order 4 or 2.

Are the first 5 subgroups of a group cyclic?

So for the first 5 subgroups listed, they are generated by ⟨ 1 ⟩, ⟨ − 1 ⟩, ⟨ i ⟩, ⟨ j ⟩, ⟨ k ⟩ and as such are cyclic. Is this correct? And is there a methodical way for identifying the subgroups? Looks correct. By Lagrange’s Theorem, a consequence of Cauchy’s Theorem the order of a subgroup must divide the order of the group.

Is the quaternion group Hamiltonian or non-abelian?

The quaternion group has the unusual property of being Hamiltonian: every subgroup of Q 8 is a normal subgroup, but the group is non-abelian. Every Hamiltonian group contains a copy of Q 8. The quaternion group Q 8 is one of the two smallest examples of a nilpotent non-abelian group, the other being the dihedral group D 4 of order 8.

Is every abelian subgroup cyclic or cyclic?

The generalized quaternion groups have the property that every abelian subgroup is cyclic. It can be shown that a finite p-group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above.