2.3 Submagmas, submonoids and subgroups
Suppose \(M\) is a magma and \(N\subseteq M\). We say that \(N\) is an submagma of \(M\) if \(N\) is closed under the binary operation on \(M\). In other words, \(N\) is a submagma if, for all \(x,y\in N\), \(xy\in N\).
Suppose \(M\) is a monoid with identity element \(1\) and \(N\subseteq M\). We say \(N\) is a submonoid if \(N\) is a submagma of \(M\) and \(1\in N\).
Suppose \(M\) is a group and \(N\subseteq M\). We say that \(N\) is a subgroup of \(M\) and write \(N\leq M\) if \(N\) is a submonoid of \(M\), which is closed under the operation of taking inverses. In other words, \(N\) is a subgroup if \(N\) is a submonoid of \(M\) and, for all \(x\in N\), \(x^{-1}\in N\).
If \(G\) is a group and \(H \subseteq G\), we say that \(H\) is a proper subgroup of \(G\) and write \(H {\lt} G\) if \(H\leq G\) but \(H \neq G\).
Suppose \(G\) is a group and \(H\subseteq G\). Then \(H \leq G\) if and only if the following three properties hold:
\(1\in H\).
\(x,y\in H\Rightarrow xy\in H\).
\(x\in H\Rightarrow x^{-1}\in H\).
Exercise. Just unravel the defintions in Defintion 2.29.
Suppose \(G\) is a group and \(H\subseteq G\). Then \(H \leq G\) if and only if the following two properties hold:
\(H\neq \emptyset \).
\(x,y\in H\Rightarrow xy^{-1}\in H\).
(\(\Rightarrow \)) Assume that \(H \leq G\). Then 1 holds because \(1 \in H\). Moreover, if \(x\), \(y\in H\), then \(y^{-1}\in H\) by Proposition 2.303. So \(xy^{-1}\in H\) by Proposition 2.302.
(\(\Leftarrow \)) Suppose that \(H\) satisfies Theorem 2.31 1 and 2. Since \(H \neq \emptyset \), we can find \(h\in H\). Then, by 2, \(1 = h h^{-1}\in H\). So \(H\) satisfies Proposition 2.301. Consequently, if \(x\in H\), then \(x^{-1} = 1 x^{-1}\in H\) by 2. So \(H\) satisfies Proposition 2.303. Finally, suppose \(x,y\in H\). Then \(y^{-1}\in H\) by what we’ve just seen. So, applying 2 again, we see that \(xy = x (y^{-1})^{-1}\in H\) showing that \(H\) satisfies Proposition 2.302.
Suppose \(G\) is a group with identity element \(e\). Then \(G\) and \(\{ e\} \) are both subgroups of \(G\). The subgroup \(\{ e\} \) is called the trivial subgroup of \(G\). We say that \(G\) is the trivial group if \(G = \{ e\} \).