How to represent the groups symbolically?
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What will be the group if the groups given by following descriptions were represented using symbols/notation (like $Z_p$ etc.)
1). Cyclic group of order $p^2 q$.
2). Semidirect product of cyclic group of order $q$ by a cyclic group of order $p^2$, where the action by conjugation of a generator is an automorphism of order $p$.
3). Semidirect product of cyclic group of order $q$ by a cyclic group of order $p^2$, where the action by conjugation of a generator is an automorphism of order $p^2$.
4). Semidirect product of cyclic group of order $pq$ by a cyclic group of order $p$, where the action by conjugation of a generator is an automorphism of order $p$.
5). Direct product of cyclic group of order $p$ and cyclic group of order $pq$, which is also same as direct product of elementary abelian group of order $p^2$ and cyclic group of order $q$.
Please help me with this question.
Thanks a lot in advance.
group-theory finite-groups
asked Aug 14 at 9:24
Buddhini Angelika
463
 |Â
show 1 more comment
up vote
0
down vote
favorite
What will be the group if the groups given by following descriptions were represented using symbols/notation (like $Z_p$ etc.)
1). Cyclic group of order $p^2 q$.
2). Semidirect product of cyclic group of order $q$ by a cyclic group of order $p^2$, where the action by conjugation of a generator is an automorphism of order $p$.
3). Semidirect product of cyclic group of order $q$ by a cyclic group of order $p^2$, where the action by conjugation of a generator is an automorphism of order $p^2$.
4). Semidirect product of cyclic group of order $pq$ by a cyclic group of order $p$, where the action by conjugation of a generator is an automorphism of order $p$.
5). Direct product of cyclic group of order $p$ and cyclic group of order $pq$, which is also same as direct product of elementary abelian group of order $p^2$ and cyclic group of order $q$.
Please help me with this question.
Thanks a lot in advance.
group-theory finite-groups
asked Aug 14 at 9:24
Buddhini Angelika
463
Well, did you look at your course notes and tried to figure things out?
– Mathematician 42
Aug 14 at 9:31
Yes, but can you please give the answer of this? Its bit difficult..
– Buddhini Angelika
Aug 14 at 9:47
1). $Z/p^2q Z$ and 5). $Z/pZ$ X $Z/pqZ$ which is same as $Z/p^2Z$ X $Z/qZ$
– Buddhini Angelika
Aug 14 at 10:11
And some authors prefer the notation $C_q$ or $mathbbZ_q=mathbbZ/qmathbbZ$ for the cyclic group of order $q$. For example the first question has the answer $C_p^2q$.
– Fakemistake
Aug 14 at 11:25
The direct product uses the symbol $times$ (times) which is not the letter $mathsfX$ (mathsfX)
– Fakemistake
Aug 14 at 11:26
 |Â
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
What will be the group if the groups given by following descriptions were represented using symbols/notation (like $Z_p$ etc.)
1). Cyclic group of order $p^2 q$.
2). Semidirect product of cyclic group of order $q$ by a cyclic group of order $p^2$, where the action by conjugation of a generator is an automorphism of order $p$.
3). Semidirect product of cyclic group of order $q$ by a cyclic group of order $p^2$, where the action by conjugation of a generator is an automorphism of order $p^2$.
4). Semidirect product of cyclic group of order $pq$ by a cyclic group of order $p$, where the action by conjugation of a generator is an automorphism of order $p$.
5). Direct product of cyclic group of order $p$ and cyclic group of order $pq$, which is also same as direct product of elementary abelian group of order $p^2$ and cyclic group of order $q$.
Please help me with this question.
Thanks a lot in advance.
group-theory finite-groups
asked Aug 14 at 9:24
Buddhini Angelika
463
What will be the group if the groups given by following descriptions were represented using symbols/notation (like $Z_p$ etc.)
1). Cyclic group of order $p^2 q$.
2). Semidirect product of cyclic group of order $q$ by a cyclic group of order $p^2$, where the action by conjugation of a generator is an automorphism of order $p$.
3). Semidirect product of cyclic group of order $q$ by a cyclic group of order $p^2$, where the action by conjugation of a generator is an automorphism of order $p^2$.
4). Semidirect product of cyclic group of order $pq$ by a cyclic group of order $p$, where the action by conjugation of a generator is an automorphism of order $p$.
5). Direct product of cyclic group of order $p$ and cyclic group of order $pq$, which is also same as direct product of elementary abelian group of order $p^2$ and cyclic group of order $q$.
Please help me with this question.
Thanks a lot in advance.
group-theory finite-groups
asked Aug 14 at 9:24
Buddhini Angelika
463
asked Aug 14 at 9:24
Buddhini Angelika
463
asked Aug 14 at 9:24
Buddhini Angelika
463
asked Aug 14 at 9:24
Buddhini Angelika
463
463
Well, did you look at your course notes and tried to figure things out?
– Mathematician 42
Aug 14 at 9:31
Yes, but can you please give the answer of this? Its bit difficult..
– Buddhini Angelika
Aug 14 at 9:47
1). $Z/p^2q Z$ and 5). $Z/pZ$ X $Z/pqZ$ which is same as $Z/p^2Z$ X $Z/qZ$
– Buddhini Angelika
Aug 14 at 10:11
And some authors prefer the notation $C_q$ or $mathbbZ_q=mathbbZ/qmathbbZ$ for the cyclic group of order $q$. For example the first question has the answer $C_p^2q$.
– Fakemistake
Aug 14 at 11:25
The direct product uses the symbol $times$ (times) which is not the letter $mathsfX$ (mathsfX)
– Fakemistake
Aug 14 at 11:26
 |Â
show 1 more comment
Well, did you look at your course notes and tried to figure things out?
– Mathematician 42
Aug 14 at 9:31
Yes, but can you please give the answer of this? Its bit difficult..
– Buddhini Angelika
Aug 14 at 9:47
1). $Z/p^2q Z$ and 5). $Z/pZ$ X $Z/pqZ$ which is same as $Z/p^2Z$ X $Z/qZ$
– Buddhini Angelika
Aug 14 at 10:11
And some authors prefer the notation $C_q$ or $mathbbZ_q=mathbbZ/qmathbbZ$ for the cyclic group of order $q$. For example the first question has the answer $C_p^2q$.
– Fakemistake
Aug 14 at 11:25
The direct product uses the symbol $times$ (times) which is not the letter $mathsfX$ (mathsfX)
– Fakemistake
Aug 14 at 11:26
Well, did you look at your course notes and tried to figure things out?
– Mathematician 42
Aug 14 at 9:31
Well, did you look at your course notes and tried to figure things out?
– Mathematician 42
Aug 14 at 9:31
Yes, but can you please give the answer of this? Its bit difficult..
– Buddhini Angelika
Aug 14 at 9:47
Yes, but can you please give the answer of this? Its bit difficult..
– Buddhini Angelika
Aug 14 at 9:47
1). $Z/p^2q Z$ and 5). $Z/pZ$ X $Z/pqZ$ which is same as $Z/p^2Z$ X $Z/qZ$
– Buddhini Angelika
Aug 14 at 10:11
1). $Z/p^2q Z$ and 5). $Z/pZ$ X $Z/pqZ$ which is same as $Z/p^2Z$ X $Z/qZ$
– Buddhini Angelika
Aug 14 at 10:11
And some authors prefer the notation $C_q$ or $mathbbZ_q=mathbbZ/qmathbbZ$ for the cyclic group of order $q$. For example the first question has the answer $C_p^2q$.
– Fakemistake
Aug 14 at 11:25
And some authors prefer the notation $C_q$ or $mathbbZ_q=mathbbZ/qmathbbZ$ for the cyclic group of order $q$. For example the first question has the answer $C_p^2q$.
– Fakemistake
Aug 14 at 11:25
The direct product uses the symbol $times$ (times) which is not the letter $mathsfX$ (mathsfX)
– Fakemistake
Aug 14 at 11:26
The direct product uses the symbol $times$ (times) which is not the letter $mathsfX$ (mathsfX)
– Fakemistake
Aug 14 at 11:26
 |Â
show 1 more comment
1 Answer
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Cyclic groups of order $n$ are often denoted multiplicatively by $C_n$. For $n=p,p^2,p^2q$ and so on we write $C_p,C_p^2,C_p^2q$. The semidirect product then is denoted by
$$
C_qrtimes_thetaC_p^2.
$$
Here the automorphism is denoted by $theta$.
answered Aug 14 at 11:43
Dietrich Burde
74.8k64185
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Cyclic groups of order $n$ are often denoted multiplicatively by $C_n$. For $n=p,p^2,p^2q$ and so on we write $C_p,C_p^2,C_p^2q$. The semidirect product then is denoted by
$$
C_qrtimes_thetaC_p^2.
$$
Here the automorphism is denoted by $theta$.
answered Aug 14 at 11:43
Dietrich Burde
74.8k64185
add a comment |Â
up vote
0
down vote
Cyclic groups of order $n$ are often denoted multiplicatively by $C_n$. For $n=p,p^2,p^2q$ and so on we write $C_p,C_p^2,C_p^2q$. The semidirect product then is denoted by
$$
C_qrtimes_thetaC_p^2.
$$
Here the automorphism is denoted by $theta$.
answered Aug 14 at 11:43
Dietrich Burde
74.8k64185
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Cyclic groups of order $n$ are often denoted multiplicatively by $C_n$. For $n=p,p^2,p^2q$ and so on we write $C_p,C_p^2,C_p^2q$. The semidirect product then is denoted by
$$
C_qrtimes_thetaC_p^2.
$$
Here the automorphism is denoted by $theta$.
answered Aug 14 at 11:43
Dietrich Burde
74.8k64185
Cyclic groups of order $n$ are often denoted multiplicatively by $C_n$. For $n=p,p^2,p^2q$ and so on we write $C_p,C_p^2,C_p^2q$. The semidirect product then is denoted by
$$
C_qrtimes_thetaC_p^2.
$$
Here the automorphism is denoted by $theta$.
answered Aug 14 at 11:43
Dietrich Burde
74.8k64185
answered Aug 14 at 11:43
Dietrich Burde
74.8k64185
answered Aug 14 at 11:43
Dietrich Burde
74.8k64185
answered Aug 14 at 11:43
Dietrich Burde
74.8k64185
74.8k64185
add a comment |Â
add a comment |Â
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Well, did you look at your course notes and tried to figure things out?
– Mathematician 42
Aug 14 at 9:31
Yes, but can you please give the answer of this? Its bit difficult..
– Buddhini Angelika
Aug 14 at 9:47
1). $Z/p^2q Z$ and 5). $Z/pZ$ X $Z/pqZ$ which is same as $Z/p^2Z$ X $Z/qZ$
– Buddhini Angelika
Aug 14 at 10:11
And some authors prefer the notation $C_q$ or $mathbbZ_q=mathbbZ/qmathbbZ$ for the cyclic group of order $q$. For example the first question has the answer $C_p^2q$.
– Fakemistake
Aug 14 at 11:25
The direct product uses the symbol $times$ (times) which is not the letter $mathsfX$ (mathsfX)
– Fakemistake
Aug 14 at 11:26