1 votes 1 votes "the union of two sub-groups neednot be a sub-group".can some-body prove without using counter example ... Set Theory & Algebra discrete-mathematics group-theory set-theory&algebra engineering-mathematics set-theory + – Vicky rix asked Mar 25, 2017 Vicky rix 867 views answer comment Share Follow See 1 comment See all 1 1 comment reply LeenSharma commented Nov 28, 2017 reply Follow Share $(S_{8} , \otimes_{8})=\left ( \left \{ 1,3,5,7 \right \} , \otimes_{8}\right )$ Let two sub group is $G$ and $H$. $G=\left \{1,3 \right \} $ $H=\left \{1,5 \right \} $ $G \cup H =\left \{1,3,5 \right \} $ You can see the order of the union can't be divied by order of the group. So union of two subgroup can't be subgroup. 1 votes 1 votes Please log in or register to add a comment.
0 votes 0 votes order of subgroup divides order of group. Quixote answered Mar 25, 2017 Quixote comment Share Follow See all 0 reply Please log in or register to add a comment.
0 votes 0 votes 3Z and 2Z are subgroups of (Z,+) but the union has elements 2 and 3. But 2+3=5∉3Z∪2Z. (Z,+) is cyclic, and hence abelian, and hence all subgroups of (Z,+)are normal. But the union of these 2 subgroups is not even a subgroup. Kaluti answered Aug 24, 2017 Kaluti comment Share Follow See all 0 reply Please log in or register to add a comment.