The dihedral group $D_8$ is a specific, non-abelian group of order 8, representing the symmetries of a square. It is not isomorphic to any other distinct group of order 8; therefore, $D_8$ is isomorphic to itself as a fundamental group structure.
Understanding the Dihedral Group $D_8$
The group $D_8$ (often denoted $D_4$ in some texts, referring to the symmetries of an $n$-gon as $D_n$) is the dihedral group of order 8. This group describes the symmetries of a square, including both rotations and reflections. It has 8 elements:
- Four rotations: $0^\circ, 90^\circ, 180^\circ, 270^\circ$.
- Four reflections: across the horizontal axis, vertical axis, and two diagonal axes.
The Landscape of Groups of Order 8
In abstract algebra, groups are classified based on their structure. For a given order (number of elements), there can be multiple non-isomorphic groups. For order 8, there are precisely five distinct groups:
- Cyclic Group of Order 8 ($C_8$ or $\mathbb{Z}_8$): An abelian group generated by a single element. It has elements $e, a, a^2, \dots, a^7$ where $a^8 = e$.
- Direct Product of Cyclic Groups ($C_4 \times C_2$ or $\mathbb{Z}_4 \times \mathbb{Z}_2$): An abelian group consisting of pairs $(x,y)$ where $x \in C_4$ and $y \in C_2$.
- Direct Product of Cyclic Groups ($C_2 \times C_2 \times C_2$ or $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$): An abelian group, often called the elementary abelian group of order 8, where every non-identity element has order 2.
- Dihedral Group of Order 8 ($D_8$ or $D_4$): A non-abelian group that describes the symmetries of a square.
- Quaternion Group ($Q_8$): A non-abelian group with unique properties, often represented by elements ${\pm 1, \pm i, \pm j, \pm k}$.
The crucial point is that $D_8$ is structurally distinct from the other four groups of order 8.
Distinguishing Properties of $D_8$
$D_8$ is non-abelian, meaning that the order of multiplication matters (e.g., $ab \neq ba$ for some elements $a,b$). This immediately differentiates it from the three abelian groups of order 8 ($C_8$, $C_4 \times C_2$, $C_2 \times C_2 \times C_2$).
To distinguish $D_8$ from the other non-abelian group of order 8, the Quaternion Group ($Q_8$), we can examine their properties:
- Number of elements of order 2:
- $D_8$ has five elements of order 2 (four reflections and the $180^\circ$ rotation).
- $Q_8$ has only one element of order 2 (the element -1).
- Center of the group: The center of a group $Z(G)$ consists of elements that commute with every other element in the group.
- The center of $D_8$ is the group consisting of the identity element and the $180^\circ$ rotation ($Z(D_8) = {1, r^2}$, where $r$ is the $90^\circ$ rotation).
- The center of $Q_8$ is ${1, -1}$.
- Inner Automorphisms: The group of inner automorphisms of $D_8$ is a group of order 4. This group is isomorphic to the quotient group $D_8/Z(D_8)$, which is a non-cyclic group of order 4, specifically the Klein four group (also known as $V_4$ or $C_2 \times C_2$).
These distinctions confirm that $D_8$ is not isomorphic to $Q_8$.
Summary Table of Groups of Order 8
| Group Type | Order | Abelian/Non-Abelian | Elements of Order 2 | Center $Z(G)$ | Notes |
|---|---|---|---|---|---|
| Cyclic Group ($C_8$) | 8 | Abelian | 1 | $C_8$ | Has an element of order 8 |
| Direct Product ($C_4 \times C_2$) | 8 | Abelian | 3 | $C_4 \times C_2$ | No element of order 8 |
| Direct Product ($C_2 \times C_2 \times C_2$) | 8 | Abelian | 7 | $C_2 \times C_2 \times C_2$ | All non-identity elements have order 2 |
| Dihedral Group ($D_8$) | 8 | Non-Abelian | 5 | ${1, r^2}$ (180° rotation) | Symmetries of a square |
| Quaternion Group ($Q_8$) | 8 | Non-Abelian | 1 | ${1, -1}$ | Unique, non-dihedral non-abelian group of order 8 |
In conclusion, $D_8$ occupies a unique position among groups of order 8 due to its specific set of properties, including its non-abelian nature, the number of elements of certain orders, and the structure of its center and inner automorphism group.
