Answer to Question #118526 in Abstract Algebra for Jflows

Question

Let A and B be two subgroups of a group G. Prove that A
cup B is a subgroup of G iff_____.
a.A\\(\\subseteq\\) B
b.B\\(\\subseteq\\) A
c.A\\(\\subseteq\\) B and B\\(\\subseteq\\) A
d.A\\(\\subseteq\\) B or B

Answer 1

Let$A\not\subset B$ and $B\not\subset A$, then $A\setminus B\neq\varnothing$ and $B\setminus A\neq\varnothing$.

Let $x\in A\setminus B$ and $y\in B\setminus A$. Prove that $xy\not\in A\cup B$, that is $xy\not\in A$ and $xy\not\in B$.

Indeed, if $xy\in A$, then $y=x^{-1}(xy)\in A$, but we choose $y\not\in A$. Similarly we obtain that $xy\not\in B$.

So we obtain that $x,y\in A\cup B$, but $xy\not\in A\cup B$, so $A\cup B$ is not a subgroup of $G$.

Therefore if $A\cup B$ is a subgroup of $G$, then $A\subset B$ or $B\subset A$.

And if $A\subset B$ or $B\subset A$, then $A\cup B=B$ or $A\cup B=A$. In every case $A\cup B$ is a subgroup of $G$.