Definition:Join of Subgroups
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Definition
Let $\struct {G, \circ}$ be a group.
Let $A$ and $B$ be subgroups of $G$.
The join of $A$ and $B$ is written and defined as:
- $A \vee B := \gen {A \cup B}$
where $\gen {A \cup B}$ is the subgroup generated by $A \cup B$.
By the definition of subgroup generator, this can alternatively be written:
- $\ds A \vee B := \bigcap \set {T: T \text { is a subgroup of } G: A \cup B \subseteq T}$
General Definition
Let $H_1, H_2, \ldots, H_n$ be subgroups of $G$.
Then the join of $H_1, H_2, \ldots, H_n$ is defined as:
- $\ds \bigvee_{k \mathop = 1}^n H_k := \gen {\bigcup_{k \mathop = 1}^n H_k}$
or:
- $\ds \bigvee_{k \mathop = 1}^n H_k := \bigcap \set {T: T \text { is a subgroup of } G: \bigcup_{k \mathop = 1}^n H_k \subseteq T}$
Also see
- Join of Subgroups is Group Generated by Union where this construction is justified.
- Union of Subgroups, where it is shown that $A \vee B = A \cup B$ iff $A \subseteq B$ or $B \subseteq A$.
Sources
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$
- 1974: Thomas W. Hungerford: Algebra ... (previous): $\S 1.2$