The **component group** of a Sato-Tate group $G$ is the quotient $G/G^0$ of $G$ by its identity component $G^0$, which is a normal subgroup of finite index.

If $G$ is the Sato-Tate group of a motive defined over a number field $K$, then the component group $G/G^0$ is canonically isomorphic to the Galois group of a finite extension $L/K$. For the motive attached to an abelian variety $A$ of dimension at most 3, $L$ is the smallest field over which all endomorphisms of $A_{\overline{K}}$ (the base change of $A$ to an algebraic closure of $K$) are defined.

Component groups are named according to their isomorphism type. This does not determine them uniquely; one can obtain a more explicit description by examining the list of generators. Common notations used in component group names include:

- $C_n$, the cyclic group of order $n$;
- $D_n$, the dihedral group of order $2n$;
- $A_n$, the alternating group on $n$ letters;
- $S_n$, the symmetric group on $n$ letters.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2021-01-01 15:15:08

**Referred to by:**

- st_group.1.4.F.48.48a.bottom
- st_group.components
- st_group.connected
- st_group.generators
- st_group.identity_component
- st_group.index
- st_group.label
- st_group.subsupgroups
- lmfdb/sato_tate_groups/main.py (line 405)
- lmfdb/sato_tate_groups/main.py (line 607)
- lmfdb/sato_tate_groups/main.py (line 1127)
- lmfdb/sato_tate_groups/main.py (line 1201)
- lmfdb/sato_tate_groups/main.py (line 1224)
- lmfdb/sato_tate_groups/templates/st_display.html (line 29)

**History:**(expand/hide all)

- 2021-01-01 15:15:08 by Andrew Sutherland (Reviewed)
- 2021-01-01 15:14:40 by Andrew Sutherland
- 2021-01-01 15:14:21 by Andrew Sutherland
- 2021-01-01 15:07:25 by Andrew Sutherland
- 2018-06-20 04:10:59 by Kiran S. Kedlaya (Reviewed)

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