Generalized quaternion group:Q16: Difference between revisions

Revision as of 23:39, 25 May 2009

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

This particular group is a finite group of order: 16

Definition

The group $Q_{16}$ , sometimes termed the generalized quaternion group' of order $16$ , is a generalized quaternion group. It can be described by the following presentation:

$G : = ⟨ a, b, c ∣ a^{4} = b^{2} = c^{2} = a b c ⟩$ .

Note that $c = a b = b a^{- 1}$ from these relations, and $b a b^{- 1} = a^{- 1}$ . This in turn forces that $b^{2} = b (b^{2}) b^{- 1} = b a^{4} b^{- 1} = a^{- 4} = b^{- 2}$ , forcing $b^{2} = a^{4} = c^{2} = a b c$ to have order two. We shall dente this element of order two, which is clearly central, as $z$ .

Subgroups

Further information: Subgroup structure of generalized quaternion group:Q16

The trivial subgroup. Isomorphic to trivial group. (1)
The center, which is a subgroup of order two, generated by $z = a^{4} = b^{2} = c^{2}$ . Isomorphic to cyclic group:Z2. (1)
The cyclic subgroup of order four generated by $a^{2}$ . Isomorphic to cyclic group:Z4. (1)
The four cyclic subgroups of order four, namely: $⟨ b ⟩$ , $⟨ a b ⟩$ , $⟨ a^{2} b ⟩$ and $⟨ a^{3} b ⟩$ . These come in two conjugacy classes of 2-subnormal subgroups, one conjugacy class comprising $⟨ a b ⟩$ and $⟨ a^{3} b ⟩$ and the other comprising $⟨ b ⟩$ and $⟨ a^{2} b ⟩$ . Isomorphic to cyclic group:Z4. (4)
The cyclic subgroup of order eight, generated by $a$ . This is characteristic; in fact, it equals the centralizer of commutator subgroup. Isomorphic to cyclic group:Z8. (1)
Two quaternion groups of order eight, namely $⟨ a^{2}, b ⟩$ and $⟨ a^{2}, a b ⟩$ . Isomorphic to quaternion group. (2)
The whole group. (1)

GAP implementation

Group ID

The generalized quaternion group of order $16$ has ID $9$ . In other words, it can be described using the SmallGroup function as:

SmallGroup(16,9)

@@ Line 16: / Line 16: @@
 # The center, which is a subgroup of order two, generated by <math>z = a^4 = b^2 = c^2</math>. Isomorphic to [[subgroup::cyclic group:Z2]]. (1)
 # The cyclic subgroup of order four generated by <math>a^2</math>. Isomorphic to [[subgroup::cyclic group:Z4]]. (1)
-# The four cyclic subgroups of order four, namely: <math>\langle b \rangle</math>, <math>\langle ab \rangle</math>, <math>\langle a^2b \rangle</math> and <math>\langle a^3b\rangle</math>. These come in two conjugacy classes of [[2-subnormal subgroup]]s, one conjugacy class comprising <math>\langle ab \rangle</math> and <math>\langle a^3b \rangle</math> and the other comprising <math>\langle b \rangle</math> and <math>\langle a^2b \rangle</math>. isomorphic to [[subgroup::cyclic group:Z4]]. (4)
+# The four cyclic subgroups of order four, namely: <math>\langle b \rangle</math>, <math>\langle ab \rangle</math>, <math>\langle a^2b \rangle</math> and <math>\langle a^3b\rangle</math>. These come in two conjugacy classes of [[2-subnormal subgroup]]s, one conjugacy class comprising <math>\langle ab \rangle</math> and <math>\langle a^3b \rangle</math> and the other comprising <math>\langle b \rangle</math> and <math>\langle a^2b \rangle</math>. Isomorphic to [[subgroup::cyclic group:Z4]]. (4)
 # The cyclic subgroup of order eight, generated by <math>a</math>. This is characteristic; in fact, it equals the [[centralizer of commutator subgroup]]. Isomorphic to [[subgroup::cyclic group:Z8]]. (1)
 # Two [[quaternion group]]s of order eight, namely <math>\langle a^2,b \rangle</math> and <math>\langle a^2, ab \rangle</math>. Isomorphic to [[subgroup::quaternion group]]. (2)