For questions about groups where the underlying set has infinite cardinality.

A group is an ordered pair $(G, \ast)$ where $G$ is a set and $\ast$ is a binary operation on $G$ such that

  • $\ast$ is associative (i.e. $(g\ast h)\ast k = g\ast(h\ast k)$ for all $g, h, k \in G$),
  • there is an element $e \in G$ such that $g\ast e = e\ast g = g$ for all $g \in G$, and
  • for every $g \in G$, there is $h \in G$ such that $g\ast h = h\ast g = e$.

The element in the second condition is called an identity element and can be shown to be unique. The element $h$ in the third condition can also be shown to be unique and is called the inverse of $g$, denoted $g^{-1}$.

An infinite group is a group $(G, \ast)$ such that the cardinality of $G$ is infinite. Some common examples include $(\mathbb{Z}, +)$, $(\mathbb{R}^*, \times)$, and $GL(n, \mathbb{R})$ with matrix multiplication.

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