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I am reading Paolo Aluffi's book, Algebra: Chapter 0 ... I am currently focused on Chapter 4, Section 3: Composition Series and Solvability ...
I need help with an aspect of Aluffi's proof of the Jordan-Holder Theorem (Theorem 3.2, page 206) which reads as follows:
Theorem 3.2 and the early part of the proof read as follows:View attachment 4912
etc ... etc
In the above proof we read:
" ... ... We may assume $$G_1 \neq {G'}_1$$. Note that $$G_1 {G'}_1 = G$$: indeed, $$G_1 {G'}_1$$ is normal in $$G$$ and $$G_1 \subset G_1 {G'}_1$$ ... ... "
Question 1
Why does it follow from $$G_1 \neq {G'}_1$$ that $$G_1 {G'}_1 = G$$ ... ... ?Question 2
Further, how does it follow that $$G_1 {G'}_1$$ is normal in $$G$$ ... ?Question 3
Further yet, how does it follow that $$G_1 \subset G_1 {G'}_1$$ ... ... ?
I hope that someone can help ...
Peter
I need help with an aspect of Aluffi's proof of the Jordan-Holder Theorem (Theorem 3.2, page 206) which reads as follows:
Theorem 3.2 and the early part of the proof read as follows:View attachment 4912
etc ... etc
In the above proof we read:
" ... ... We may assume $$G_1 \neq {G'}_1$$. Note that $$G_1 {G'}_1 = G$$: indeed, $$G_1 {G'}_1$$ is normal in $$G$$ and $$G_1 \subset G_1 {G'}_1$$ ... ... "
Question 1
Why does it follow from $$G_1 \neq {G'}_1$$ that $$G_1 {G'}_1 = G$$ ... ... ?Question 2
Further, how does it follow that $$G_1 {G'}_1$$ is normal in $$G$$ ... ?Question 3
Further yet, how does it follow that $$G_1 \subset G_1 {G'}_1$$ ... ... ?
I hope that someone can help ...
Peter