nontrivial solvable group contains a nontrivial characteristic abelian subgroup












1












$begingroup$


I am reading about algebra and I have these questions to prove along with my attempts, any help is highly appreciated:



(i) Show that a nontrivial solvable group contains a nontrivial characteristic abelian subgroup.



(ii) Show that a nontrivial solvable group has a nontrivial abelian factor subgroup.



(iii) Show that a nonsolvable group contains a nontrivial characteristic subgroup $N$ such that $ N'= N $



(iv) Let $G$ be a solvable group, and let $ 1 neq H triangleleft G $. Show that there is an $A$ such that $1 neq A triangleleft G $ and $A leq H $ and $A$ is abelian.



My attempt:



If a group is solvable, then there must be a $G^{(s)} = 1$ for some integer $s geq 0$. Then there must be a subnormal series of $G$ with abelian factors, which solves the second question( right ?).



If $G$ is not solvable, its derived series cannot reach $1$.










share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    Note that the derived series is not just a subnormal series, but a normal series and even a characteristic series. This should help.
    $endgroup$
    – verret
    Jan 9 at 5:04










  • $begingroup$
    For i) you can also take the socle, i.e. the product of all minimal normal subgroups.
    $endgroup$
    – the_fox
    Jan 9 at 6:30










  • $begingroup$
    @the_fox Not all groups have minimal normal subgroups - $({mathbb Z},+)$ for example.
    $endgroup$
    – Derek Holt
    Jan 9 at 8:24










  • $begingroup$
    @DerekHolt There was a "finite groups" tag, which made me think the question was about finite groups exclusively.
    $endgroup$
    – the_fox
    Jan 9 at 9:14










  • $begingroup$
    (iii) is false for suitable infinite groups. (i),(ii),(iv) are true without finiteness assumption.
    $endgroup$
    – YCor
    Jan 11 at 2:17
















1












$begingroup$


I am reading about algebra and I have these questions to prove along with my attempts, any help is highly appreciated:



(i) Show that a nontrivial solvable group contains a nontrivial characteristic abelian subgroup.



(ii) Show that a nontrivial solvable group has a nontrivial abelian factor subgroup.



(iii) Show that a nonsolvable group contains a nontrivial characteristic subgroup $N$ such that $ N'= N $



(iv) Let $G$ be a solvable group, and let $ 1 neq H triangleleft G $. Show that there is an $A$ such that $1 neq A triangleleft G $ and $A leq H $ and $A$ is abelian.



My attempt:



If a group is solvable, then there must be a $G^{(s)} = 1$ for some integer $s geq 0$. Then there must be a subnormal series of $G$ with abelian factors, which solves the second question( right ?).



If $G$ is not solvable, its derived series cannot reach $1$.










share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    Note that the derived series is not just a subnormal series, but a normal series and even a characteristic series. This should help.
    $endgroup$
    – verret
    Jan 9 at 5:04










  • $begingroup$
    For i) you can also take the socle, i.e. the product of all minimal normal subgroups.
    $endgroup$
    – the_fox
    Jan 9 at 6:30










  • $begingroup$
    @the_fox Not all groups have minimal normal subgroups - $({mathbb Z},+)$ for example.
    $endgroup$
    – Derek Holt
    Jan 9 at 8:24










  • $begingroup$
    @DerekHolt There was a "finite groups" tag, which made me think the question was about finite groups exclusively.
    $endgroup$
    – the_fox
    Jan 9 at 9:14










  • $begingroup$
    (iii) is false for suitable infinite groups. (i),(ii),(iv) are true without finiteness assumption.
    $endgroup$
    – YCor
    Jan 11 at 2:17














1












1








1





$begingroup$


I am reading about algebra and I have these questions to prove along with my attempts, any help is highly appreciated:



(i) Show that a nontrivial solvable group contains a nontrivial characteristic abelian subgroup.



(ii) Show that a nontrivial solvable group has a nontrivial abelian factor subgroup.



(iii) Show that a nonsolvable group contains a nontrivial characteristic subgroup $N$ such that $ N'= N $



(iv) Let $G$ be a solvable group, and let $ 1 neq H triangleleft G $. Show that there is an $A$ such that $1 neq A triangleleft G $ and $A leq H $ and $A$ is abelian.



My attempt:



If a group is solvable, then there must be a $G^{(s)} = 1$ for some integer $s geq 0$. Then there must be a subnormal series of $G$ with abelian factors, which solves the second question( right ?).



If $G$ is not solvable, its derived series cannot reach $1$.










share|cite|improve this question











$endgroup$




I am reading about algebra and I have these questions to prove along with my attempts, any help is highly appreciated:



(i) Show that a nontrivial solvable group contains a nontrivial characteristic abelian subgroup.



(ii) Show that a nontrivial solvable group has a nontrivial abelian factor subgroup.



(iii) Show that a nonsolvable group contains a nontrivial characteristic subgroup $N$ such that $ N'= N $



(iv) Let $G$ be a solvable group, and let $ 1 neq H triangleleft G $. Show that there is an $A$ such that $1 neq A triangleleft G $ and $A leq H $ and $A$ is abelian.



My attempt:



If a group is solvable, then there must be a $G^{(s)} = 1$ for some integer $s geq 0$. Then there must be a subnormal series of $G$ with abelian factors, which solves the second question( right ?).



If $G$ is not solvable, its derived series cannot reach $1$.







abstract-algebra group-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 9 at 8:25









Derek Holt

53.8k53571




53.8k53571










asked Jan 9 at 3:23









NawalNawal

223




223








  • 2




    $begingroup$
    Note that the derived series is not just a subnormal series, but a normal series and even a characteristic series. This should help.
    $endgroup$
    – verret
    Jan 9 at 5:04










  • $begingroup$
    For i) you can also take the socle, i.e. the product of all minimal normal subgroups.
    $endgroup$
    – the_fox
    Jan 9 at 6:30










  • $begingroup$
    @the_fox Not all groups have minimal normal subgroups - $({mathbb Z},+)$ for example.
    $endgroup$
    – Derek Holt
    Jan 9 at 8:24










  • $begingroup$
    @DerekHolt There was a "finite groups" tag, which made me think the question was about finite groups exclusively.
    $endgroup$
    – the_fox
    Jan 9 at 9:14










  • $begingroup$
    (iii) is false for suitable infinite groups. (i),(ii),(iv) are true without finiteness assumption.
    $endgroup$
    – YCor
    Jan 11 at 2:17














  • 2




    $begingroup$
    Note that the derived series is not just a subnormal series, but a normal series and even a characteristic series. This should help.
    $endgroup$
    – verret
    Jan 9 at 5:04










  • $begingroup$
    For i) you can also take the socle, i.e. the product of all minimal normal subgroups.
    $endgroup$
    – the_fox
    Jan 9 at 6:30










  • $begingroup$
    @the_fox Not all groups have minimal normal subgroups - $({mathbb Z},+)$ for example.
    $endgroup$
    – Derek Holt
    Jan 9 at 8:24










  • $begingroup$
    @DerekHolt There was a "finite groups" tag, which made me think the question was about finite groups exclusively.
    $endgroup$
    – the_fox
    Jan 9 at 9:14










  • $begingroup$
    (iii) is false for suitable infinite groups. (i),(ii),(iv) are true without finiteness assumption.
    $endgroup$
    – YCor
    Jan 11 at 2:17








2




2




$begingroup$
Note that the derived series is not just a subnormal series, but a normal series and even a characteristic series. This should help.
$endgroup$
– verret
Jan 9 at 5:04




$begingroup$
Note that the derived series is not just a subnormal series, but a normal series and even a characteristic series. This should help.
$endgroup$
– verret
Jan 9 at 5:04












$begingroup$
For i) you can also take the socle, i.e. the product of all minimal normal subgroups.
$endgroup$
– the_fox
Jan 9 at 6:30




$begingroup$
For i) you can also take the socle, i.e. the product of all minimal normal subgroups.
$endgroup$
– the_fox
Jan 9 at 6:30












$begingroup$
@the_fox Not all groups have minimal normal subgroups - $({mathbb Z},+)$ for example.
$endgroup$
– Derek Holt
Jan 9 at 8:24




$begingroup$
@the_fox Not all groups have minimal normal subgroups - $({mathbb Z},+)$ for example.
$endgroup$
– Derek Holt
Jan 9 at 8:24












$begingroup$
@DerekHolt There was a "finite groups" tag, which made me think the question was about finite groups exclusively.
$endgroup$
– the_fox
Jan 9 at 9:14




$begingroup$
@DerekHolt There was a "finite groups" tag, which made me think the question was about finite groups exclusively.
$endgroup$
– the_fox
Jan 9 at 9:14












$begingroup$
(iii) is false for suitable infinite groups. (i),(ii),(iv) are true without finiteness assumption.
$endgroup$
– YCor
Jan 11 at 2:17




$begingroup$
(iii) is false for suitable infinite groups. (i),(ii),(iv) are true without finiteness assumption.
$endgroup$
– YCor
Jan 11 at 2:17










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