nontrivial solvable group contains a nontrivial characteristic abelian subgroup
$begingroup$
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 $1$.
abstract-algebra group-theory
edited Jan 9 at 8:25
Derek Holt
53.8k53571
asked Jan 9 at 3:23
NawalNawal
223
$endgroup$
add a comment |
$begingroup$
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 $1$.
abstract-algebra group-theory
edited Jan 9 at 8:25
Derek Holt
53.8k53571
asked Jan 9 at 3:23
NawalNawal
223
$endgroup$
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
add a comment |
$begingroup$
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 $1$.
abstract-algebra group-theory
edited Jan 9 at 8:25
Derek Holt
53.8k53571
asked Jan 9 at 3:23
NawalNawal
223
$endgroup$
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
abstract-algebra group-theory
edited Jan 9 at 8:25
Derek Holt
53.8k53571
asked Jan 9 at 3:23
NawalNawal
223
edited Jan 9 at 8:25
Derek Holt
53.8k53571
asked Jan 9 at 3:23
NawalNawal
223
edited Jan 9 at 8:25
Derek Holt
53.8k53571
edited Jan 9 at 8:25
Derek Holt
53.8k53571
edited Jan 9 at 8:25
Derek Holt
53.8k53571
53.8k53571
asked Jan 9 at 3:23
NawalNawal
223
asked Jan 9 at 3:23
NawalNawal
223
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
add a comment |
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
add a comment |
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$begingroup$
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