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I have made two posts recently concerning the composition series of groups and have received considerable help from Euge and Deveno regarding this topic ... in particular, Euge and Deveno have pointed out the role of the Correspondence Theorem for Groups (Lattice Isomorphism Theorem for Groups) in analysing composition series ...
I am trying to fully understand the role of the Correspondence Theorem for Groupsin analysing composition series ... but need a bit more help ...
The issue I am focused on is the following:
Aluffi in his book, Algebra: Chapter 0 in the proof of the Jordan-Holder Theorem (pages 206 - 207) ... given a composition series:
$$G = G_0 \supsetneq G_1 \supsetneq G_2 \supsetneq \ ... \ ... \ \supsetneq G_n = \{e \} $$
states the following:
" ... ... there are no proper normal subgroups between $$G_1$$ and $$G$$ since $$G/G_1$$ is simple ... ... "
... so then more generally we have the following:
... there are no proper normal subgroups between $$G_{ i + 1}$$ and $$G_i$$ since $$G_i/G_{ i + 1}$$ is simple ... ... "Now ... ... in a previous post, Euge pointed out that this statement can be established through applying the Correspondence Theorem ... but how, exactly?To establish a notation, I am providing the statement of the Correspondence Theorem from Joseph J Rotman's undergraduate text, An Introduction to Abstract Algebra with Applications (Third Edition) ... as follows ... :View attachment 4919Now to restate the above in terms of our problem, we have:
$$G_{i + 1} \triangleleft G_i $$
Then $$\text{ Sub}( G_i ; G_{i + 1} )$$ is the family of all those subgroups $$S$$ of $$G_i$$ containing $$G_{i + 1}$$
and
$$\text{ Sub}( G_i / G_{i + 1} )$$ is the family of all subgroups of $$ G_i / G_{i + 1} $$
Now ... we need to show that
$$G_i / G_{i + 1}$$ is simple $$\Longrightarrow$$ there are no proper normal subgroups between $$G_i$$ and $$G_{i + 1}$$
... BUT ... ... how exactly do we do this ... ... ?Seems like we should use the Correspondence Theorem part (iii) ... but how exactly ... ?
Hope someone can help ...
Peter
I am trying to fully understand the role of the Correspondence Theorem for Groupsin analysing composition series ... but need a bit more help ...
The issue I am focused on is the following:
Aluffi in his book, Algebra: Chapter 0 in the proof of the Jordan-Holder Theorem (pages 206 - 207) ... given a composition series:
$$G = G_0 \supsetneq G_1 \supsetneq G_2 \supsetneq \ ... \ ... \ \supsetneq G_n = \{e \} $$
states the following:
" ... ... there are no proper normal subgroups between $$G_1$$ and $$G$$ since $$G/G_1$$ is simple ... ... "
... so then more generally we have the following:
... there are no proper normal subgroups between $$G_{ i + 1}$$ and $$G_i$$ since $$G_i/G_{ i + 1}$$ is simple ... ... "Now ... ... in a previous post, Euge pointed out that this statement can be established through applying the Correspondence Theorem ... but how, exactly?To establish a notation, I am providing the statement of the Correspondence Theorem from Joseph J Rotman's undergraduate text, An Introduction to Abstract Algebra with Applications (Third Edition) ... as follows ... :View attachment 4919Now to restate the above in terms of our problem, we have:
$$G_{i + 1} \triangleleft G_i $$
Then $$\text{ Sub}( G_i ; G_{i + 1} )$$ is the family of all those subgroups $$S$$ of $$G_i$$ containing $$G_{i + 1}$$
and
$$\text{ Sub}( G_i / G_{i + 1} )$$ is the family of all subgroups of $$ G_i / G_{i + 1} $$
Now ... we need to show that
$$G_i / G_{i + 1}$$ is simple $$\Longrightarrow$$ there are no proper normal subgroups between $$G_i$$ and $$G_{i + 1}$$
... BUT ... ... how exactly do we do this ... ... ?Seems like we should use the Correspondence Theorem part (iii) ... but how exactly ... ?
Hope someone can help ...
Peter