Groups: Standing Notation

(i) Base. This setting extends Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity: its conventions and notation are in force.

(ii) Groups and monoids. For a group $(G,\cdot)$, the product of $g,h\in G$ is written $gh$, the identity element is $e$ (or $e_{G}$), the inverse of $g$ is $g^{-1}$, and for $n\in\mathbb{Z}$ the power of $g$ is $g^{n}$. A group is named by its underlying set when the operation is clear. A monoid $(M,\cdot)$ is written the same way, with identity $e$, and the inverse of a unit $u$ of $M$, unique by Units of a Monoid: Unique Inverses, Products, and the Group of Units (i), is $u^{-1}$.

(iii) Additive notation. For an abelian group one may write instead $g+h$ for the product of $g$ and $h$, $0$ for the identity, $-g$ for the inverse of $g$, and $ng$ for the power $g^{n}$ ($n\in\mathbb{Z}$).

(iv) Subgroups and cosets. $H\le G$ means that $H$ is a subgroup of $G$. For $H\le G$ and $g\in G$, $gH$ is the left coset and $Hg$ the right coset of $H$ by $g$, $G/H$ is the set of left cosets of $H$, $|G|$ is the order of $G$ when $G$ is finite, and $[G:H]$ is the index of $H$ in $G$.

(v) Homomorphisms. For a group homomorphism $f:G\to H$, $\ker f$ is its kernel and $f(G)$ its image. $G\cong H$ means that there is an isomorphism from $G$ to $H$.

(vi) Generated sets and orders of elements. For $g\in G$, $\langle g\rangle$ is the set generated by $g$ and, when $g$ has finite order, $\operatorname{ord}(g)$ is the order of $g$.

(vii) Actions. For an action of $G$ on a set $X$, $g\cdot x$ is the image of $x\in X$ under $g\in G$, $Gx$ is the orbit of $x$, and $G_{x}$ is its stabilizer.