Periodic normal implies amalgam-characteristic
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., periodic normal subgroup) must also satisfy the second subgroup property (i.e., amalgam-characteristic subgroup)
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Suppose is a group and is a periodic normal subgroup of : in other words, is a normal subgroup of as well as a periodic group (every element of is of finite order). Then, is an amalgam-characteristic subgroup of : is characteristic in .
- Quotient of amalgamated free product by amalgamated normal subgroup equals free product of quotient groups
- Free product of nontrivial groups has no nontrivial periodic normal subgroup
- Normality is upper join-closed
- Normality satisfies image condition
Given: A group , a periodic normal subgroup of . .
To prove: is a characteristic subgroup in .
- By fact (1), .
- has no nontrivial periodic normal subgroup: By fact (2), if is a proper subgroup of , then has no nontrivial periodic normal subgroup. If , then is trivial and hence has no nontrivial periodic normal subgroup.
- is a finite normal subgroup of : Since is normal in both copies of , it is normal in . Also, is finite.
- is the unique largest periodic normal subgroup of : Suppose is a finite normal subgroup of . Then, by fact (4), the image of in the quotient map is a normal subgroup of . Also, since is periodic, its image is periodic. In step (2), we concluded that has no nontrivial periodic normal subgroup. Thus, the image of is trivial, so .
- is characteristic in : This follows since it is the unique largest periodic normal subgroup of .