- #1
QIsReluctant
- 37
- 3
Note: I only need help on the underlined portion of the problem, but I'm including all parts since they may provide relevant information. Thanks in advance.
1. Homework Statement
Let S be a subset of a group G such that g−1Sg ⊂ S for any g∈G. Show that the subgroup ⟨S⟩ generated by S is normal.
Let T be any subset of G and let S = g−1Tg. Show that ⟨S⟩ is the normal subgroup generated by T.
I apply the first part of the problem to see that <S> is normal, and that is as far as I am getting. I know that <S> ⊇ <T> by the definitions, but since we have such little information about T I can't get much further. If the normal/"ordinary" subgroups containing S and T were the same then the conclusion would be obvious but the definition of G seems to preclude this.
1. Homework Statement
Let S be a subset of a group G such that g−1Sg ⊂ S for any g∈G. Show that the subgroup ⟨S⟩ generated by S is normal.
Let T be any subset of G and let S = g−1Tg. Show that ⟨S⟩ is the normal subgroup generated by T.
The Attempt at a Solution
I apply the first part of the problem to see that <S> is normal, and that is as far as I am getting. I know that <S> ⊇ <T> by the definitions, but since we have such little information about T I can't get much further. If the normal/"ordinary" subgroups containing S and T were the same then the conclusion would be obvious but the definition of G seems to preclude this.