The Intersection and Union of Two Subgroups

The Intersection and Union of Two Subgroups

Consider a group $(G, *)$ and suppose that $(S, *)$ and $(T, *)$ are both subgroups of $(G, *)$. What can we then say about the set intersection $S \cap T$ and set union $S \cup T$ with respect to the operation $*$? Will they always form groups or not? The following theorems fortunately answer this question.

Theorem 1: Let $(G, *)$ be a group and $(S, *)$ and $(T, *)$ be subgroups. Then $(S \cap T, *)$ is also a subgroup.
  • Proof: Since $(S, *)$ and $(T, *)$ are subgroups of $(G, *)$ we have that $S, T \subseteq G$ and so $S \cap T \subseteq G$. We therefore only need to show that $S \cap T$ is closed under $*$ and that for each $a \in S \cap T$ there exists an $a^{-1} \in S \cap T$ such that $a * a^{-1} = e$ and $a^{-1} * a = e$ where $e \in G$ is the identity with respect to $*$.
  • Let $x, y \in S \cap T$. Then $x, y \in S$ and $x, y \in T$ and since $(S, *)$ and $(T, *)$ are groups themselves, we have that $S$ and $T$ are both closed under $*$. Therefore $(x * y) \in S$ and $(x * y) \in T$. Therefore $(x * y) \in S \cap T$, so $S \cap T$ is closed under $*$.
  • Now let $a \in S \cap T$. Then $a \in S$ and $a \in T$. Since $(S, *)$ and $(T, *)$ are groups, we have that there exists $a^{-1} \in S$ such that $a * a^{-1} = e$ and $a^{-1} * a = e$ and there exists $a^{-1'} \in T$ such that $a * a^{-1'} = e$ and $a^{-1} * a = e$. Since $S, T \subseteq G$ we must have that $a^{-1} = a^{-1'}$ otherwise this would contradict the 'existence of a unique inverse for $a \in G$. Therefore $a^{-1} \in S$ and $a^{-1} \in T$ so there exists an $a^{-1} \in S \cap T$ such that $a * a^{-1} = e$ and $a^{-1} * a = e$.
  • Therefore $(S \cap T, *)$ is a group. $\blacksquare$
Theorem 2: Let $(G, *)$ be a group and let $(S, *)$ and $(T, *)$ be subgroups. Then $(S \cup T, *)$ is a group if and only if $S \subseteq T$ or $S \supseteq T$.
  • Proof: $\Rightarrow$ Suppose that $(S \cup T, *)$ is a group. Then all of the group axioms hold on $S \cup T$ with respect to the operation $*$ and so for all $x, y, z \in S \cup T$ we have that $(x * y) \in S \cup T$; $x * (y * z) = (x * y) * z$; there exists an $e \in S \cup T$ such that $x * e = x$ and $e * x = x$; and $x \in S \cup T$ implies that there exists $x^{-1} \in S \cup T$ such that $x * x^{-1} = e$ and $x^{-1} * x = e$.
  • Now suppose that $S \not \subseteq T$ and $S \not \supseteq T$. Then there exists an $x \in S$ such that $x \not \in T$ and similarly there exists a $y \in T$ such that $y \not in S$. Clearly $x, y \in S \cup T$ so $(x * y) \in S \cup T$ since $(S \cup T, *)$ is a group (and hence closed under $*$). So either $(x * y) \in S$ or $(x * y) \in T$.
  • Suppose that $(x * y) \in S$. Since $x \in S$ and $(S, *)$ is a group, we have that $x^{-1} \in S$. Since $(S, *)$ is a group, we also have that $S$ is closed under $*$ and so:
(1)
\begin{align} \quad x^{-1} * (x * y) = (x^{-1} * x) * y = e * y = y \in S \end{align}
  • But this is a contradiction since $y \not \in S$.
  • Similarly, suppose that $(x * y) \in T$. Since $y \in T$ and $(T, *)$ is a group, we have that $y^{-1} \in T$. Since $(T, *)$ is a group, we also have that $T$ is closed under $*$ and so:
(2)
\begin{align} \quad (x * y) * y^{-1} = x * (y * y^{-1}) = x * e = x \in T \end{align}
  • But this too is a contradiction since $x \not \in T$.
  • Therefore the assumption that $S \not \subseteq T$ and $S \not \supseteq T$ was false. Therefore either $S \subseteq T$ or $S \supseteq T$.
  • $\Leftarrow$ Suppose that now $S \subseteq T$. Then $S \cup T = T$ and since $(T, *)$ is a group, we have that $(S \cup T, *)$ is a group. Now suppose that $S \supseteq T$. Then $S \cup T = S$ and since $(S, *)$ is a group, we once again have that $(S \cup T, *)$ is a group. $\blacksquare$
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