How many noncyclic submodules with $9$ elements does $V$ have?
$begingroup$
Let $V=mathbb{Z}/(3) oplus mathbb{Z}/(3) oplus mathbb{Z}/(9)$.
- How many submodules with $3$ elements does $V$ have?
Because $phi(3)=2$, each subgroup of order $3$ has $2$ elements of order $3$. Since no $2$ cyclic subgroup can have an element of order $3$ in common, there $6div 2=3$ cyclic subgroups. Note that there are no non-cyclic subgroups with order $3$ because $mathbb{Z}_3$ is always cyclic.
- How many of the submodules $W$ of $V$ with $3$ elements have a
complementary direct summand, i.e., are such that there exists a
submodule $W'$ of $V$ with $V=Woplus W'$?
I want to say that $W'=mathbb{Z}_3oplus mathbb{Z}_9$ but it feels like I am guessing. Also how can I determine how many submodules there are with this property?
- How many cyclic submodules with $9$ elements does $V$ have?
Because $phi(9)=6$, each subgroup of order $9$ has $6$ elements of order $9$. Since no $2$ cyclic subgroups can have an element of order $9$ in common, there are $54div 6=9$ cyclic subgroups.
- How many noncyclic submodules with $9$ elements does $V$ have?
The noncyclic subgroups with order $9$ are isomorphic to $mathbb{Z}_3oplus mathbb{Z}_3$. i.e. In a non-cyclic subgroups of order $3$ each non-identity element is of order $3$.
Where do I go from here?
abstract-algebra
asked Dec 24 '18 at 3:42
Username UnknownUsername Unknown
1,16742258
$endgroup$
add a comment |
$begingroup$
Let $V=mathbb{Z}/(3) oplus mathbb{Z}/(3) oplus mathbb{Z}/(9)$.
- How many submodules with $3$ elements does $V$ have?
Because $phi(3)=2$, each subgroup of order $3$ has $2$ elements of order $3$. Since no $2$ cyclic subgroup can have an element of order $3$ in common, there $6div 2=3$ cyclic subgroups. Note that there are no non-cyclic subgroups with order $3$ because $mathbb{Z}_3$ is always cyclic.
- How many of the submodules $W$ of $V$ with $3$ elements have a
complementary direct summand, i.e., are such that there exists a
submodule $W'$ of $V$ with $V=Woplus W'$?
I want to say that $W'=mathbb{Z}_3oplus mathbb{Z}_9$ but it feels like I am guessing. Also how can I determine how many submodules there are with this property?
- How many cyclic submodules with $9$ elements does $V$ have?
Because $phi(9)=6$, each subgroup of order $9$ has $6$ elements of order $9$. Since no $2$ cyclic subgroups can have an element of order $9$ in common, there are $54div 6=9$ cyclic subgroups.
- How many noncyclic submodules with $9$ elements does $V$ have?
The noncyclic subgroups with order $9$ are isomorphic to $mathbb{Z}_3oplus mathbb{Z}_3$. i.e. In a non-cyclic subgroups of order $3$ each non-identity element is of order $3$.
Where do I go from here?
abstract-algebra
asked Dec 24 '18 at 3:42
Username UnknownUsername Unknown
1,16742258
$endgroup$
add a comment |
$begingroup$
Let $V=mathbb{Z}/(3) oplus mathbb{Z}/(3) oplus mathbb{Z}/(9)$.
- How many submodules with $3$ elements does $V$ have?
Because $phi(3)=2$, each subgroup of order $3$ has $2$ elements of order $3$. Since no $2$ cyclic subgroup can have an element of order $3$ in common, there $6div 2=3$ cyclic subgroups. Note that there are no non-cyclic subgroups with order $3$ because $mathbb{Z}_3$ is always cyclic.
- How many of the submodules $W$ of $V$ with $3$ elements have a
complementary direct summand, i.e., are such that there exists a
submodule $W'$ of $V$ with $V=Woplus W'$?
I want to say that $W'=mathbb{Z}_3oplus mathbb{Z}_9$ but it feels like I am guessing. Also how can I determine how many submodules there are with this property?
- How many cyclic submodules with $9$ elements does $V$ have?
Because $phi(9)=6$, each subgroup of order $9$ has $6$ elements of order $9$. Since no $2$ cyclic subgroups can have an element of order $9$ in common, there are $54div 6=9$ cyclic subgroups.
- How many noncyclic submodules with $9$ elements does $V$ have?
The noncyclic subgroups with order $9$ are isomorphic to $mathbb{Z}_3oplus mathbb{Z}_3$. i.e. In a non-cyclic subgroups of order $3$ each non-identity element is of order $3$.
Where do I go from here?
abstract-algebra
asked Dec 24 '18 at 3:42
Username UnknownUsername Unknown
1,16742258
$endgroup$
Let $V=mathbb{Z}/(3) oplus mathbb{Z}/(3) oplus mathbb{Z}/(9)$.
- How many submodules with $3$ elements does $V$ have?
Because $phi(3)=2$, each subgroup of order $3$ has $2$ elements of order $3$. Since no $2$ cyclic subgroup can have an element of order $3$ in common, there $6div 2=3$ cyclic subgroups. Note that there are no non-cyclic subgroups with order $3$ because $mathbb{Z}_3$ is always cyclic.
- How many of the submodules $W$ of $V$ with $3$ elements have a
complementary direct summand, i.e., are such that there exists a
submodule $W'$ of $V$ with $V=Woplus W'$?
I want to say that $W'=mathbb{Z}_3oplus mathbb{Z}_9$ but it feels like I am guessing. Also how can I determine how many submodules there are with this property?
- How many cyclic submodules with $9$ elements does $V$ have?
Because $phi(9)=6$, each subgroup of order $9$ has $6$ elements of order $9$. Since no $2$ cyclic subgroups can have an element of order $9$ in common, there are $54div 6=9$ cyclic subgroups.
- How many noncyclic submodules with $9$ elements does $V$ have?
The noncyclic subgroups with order $9$ are isomorphic to $mathbb{Z}_3oplus mathbb{Z}_3$. i.e. In a non-cyclic subgroups of order $3$ each non-identity element is of order $3$.
Where do I go from here?
abstract-algebra
abstract-algebra
asked Dec 24 '18 at 3:42
Username UnknownUsername Unknown
1,16742258
asked Dec 24 '18 at 3:42
Username UnknownUsername Unknown
1,16742258
asked Dec 24 '18 at 3:42
Username UnknownUsername Unknown
1,16742258
asked Dec 24 '18 at 3:42
Username UnknownUsername Unknown
1,16742258
asked Dec 24 '18 at 3:42
Username UnknownUsername Unknown
1,16742258
1,16742258
add a comment |
add a comment |
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