Polam wrote: |
Suppose H and K are two subgroups of a group G, and that the orders of H and K are 12 and 30 respectively. Which of the following cannot be the order of the subgroup of G generated by H and K, and why? A. 30 B. 60 C. 120 D. 360 E. Countably infinite
Any suggestions are welcome. |
If the subgroup F of G generated by H and K is finite, then Lagrange theorem says that the o(F) is a multiple of o(H) = 12. So o(F) cannot be 30.
Polam wrote: |
By the way, I still don’t see how the order of HK can be infinite. |
The answer (E) is possible. For example, the group is countably infinite and is generated by its subgroups and . The orders of these two subgroups are 12 and 30. (In this case, HK is not a subgroup.)_________________
世上沒有完美的人完美的事，而我們的責任就是要令自己的表演達至最精彩最完美。所以魔術一直都沒有停頓下來，與時代一起進步去追求無止境的完美。