cyclic groups with the same cardinality but different generators are isomorphic. Is naming the unit...
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cyclic groups with the same cardinality but different generators are isomorphic. Is naming the unit equivalent to naming the map?
When are the product of cyclic groups also cyclic?How the symmetric groups with same cardinality can be isomorphic?When are groups with the same class equation not isomorphic?Subgroup of a cyclic groups and are isomorphicIf two groups are generated by the same number of generators, and the generators have the same orders, are the groups isomorphic?Are groups with all the same Hom sets already isomorphic?An infinite cyclic group has two generators. Does the cardinality of the infinite set matter?Prove that any two cyclic groups of the same order are isomorphic?When are same order groups are isomorphicProve that all cyclic groups of the same order are isomorphic
$begingroup$
I just got a hw back from teacher and am curious about a marked mistake.
problem (6) was to "describe all isomorphisms from $Z_5$ to $Z_5$" and to (7) "describe all isomorphisms from $Z_4$ to $Z_4$."
I wrote that $Z_5 cong$ <$bar1$> $cong$ <$bar2$> $cong$ <$bar3$> $cong$ <$bar4$> where $bar a$ = a mod 5 and "<" , ">" denote 'the group generated by'.
Similarly, I wrote that $Z_4 cong$ <$hat1$> $cong$ <$hat3$> where $hat b$ = b mod 4 and "<" , ">" denote 'the group generated by'.
Teacher gave me half points for both problems because I 'did not explicitly write the map.' Isn't writing down that two cyclic groups with different generators are isomorphic a pretty explicit map? Does it seem a little overkill?
I don't know, I realize this isn't exactly the place for the question I'm about to ask, but I am just pretty floored and don't know of another community that would be more appropriate. Does this seem like a gender thing to you? My partner, an older male, has a fairly good understanding of algebra, but is receiving considerably higher marks. (He did problems 1 -3, while I did 4-7) -- see attached. I want to go to someone at the school, because it seems pretty biased to me, but would prefer to hear all of your opinions first.
http://math.sfsu.edu/gubeladze/classes/spring2019/335/homework%204%20(2019).pdf
https://imgur.com/VzItRZ7
https://imgur.com/zYNcpay
abstract-algebra group-theory cyclic-groups
asked Mar 9 at 2:04
nessness
426
$endgroup$
|
show 1 more comment
$begingroup$
I just got a hw back from teacher and am curious about a marked mistake.
problem (6) was to "describe all isomorphisms from $Z_5$ to $Z_5$" and to (7) "describe all isomorphisms from $Z_4$ to $Z_4$."
I wrote that $Z_5 cong$ <$bar1$> $cong$ <$bar2$> $cong$ <$bar3$> $cong$ <$bar4$> where $bar a$ = a mod 5 and "<" , ">" denote 'the group generated by'.
Similarly, I wrote that $Z_4 cong$ <$hat1$> $cong$ <$hat3$> where $hat b$ = b mod 4 and "<" , ">" denote 'the group generated by'.
Teacher gave me half points for both problems because I 'did not explicitly write the map.' Isn't writing down that two cyclic groups with different generators are isomorphic a pretty explicit map? Does it seem a little overkill?
I don't know, I realize this isn't exactly the place for the question I'm about to ask, but I am just pretty floored and don't know of another community that would be more appropriate. Does this seem like a gender thing to you? My partner, an older male, has a fairly good understanding of algebra, but is receiving considerably higher marks. (He did problems 1 -3, while I did 4-7) -- see attached. I want to go to someone at the school, because it seems pretty biased to me, but would prefer to hear all of your opinions first.
http://math.sfsu.edu/gubeladze/classes/spring2019/335/homework%204%20(2019).pdf
https://imgur.com/VzItRZ7
https://imgur.com/zYNcpay
abstract-algebra group-theory cyclic-groups
asked Mar 9 at 2:04
nessness
426
$endgroup$
1
$begingroup$
The relation of being isomorphic means an isomorphism exists. That is not as specific as actually writing down an isomorphism. The point is that groups that are isomorphic have multiple isomorphisms between them (unless the groups have order 1 or 2). As a matter of notation, writing $G cong H$ is not the same as writing that a specific function $f colon G rightarrow H$ is an isomorphism. If someone learning English writes "I yesterday to store went" do you think that deserves close to full marks because it is clear what the intended meaning was despite mistakes?
$endgroup$
– KCd
2 days ago
1
$begingroup$
Now I looked at the links (I am a bit surprised you included your name at the top of the first page). I don't see a case of gender bias in the grades on these answers. You made the same error in questions 6 and 7, and got graded the same way for both of them. Your hmwk partner got 7/10 on question 1. In question 5, which you did and for which you got full marks, the task was to decide if the groups are isomorphic, not what an example of an isomorphism is. That is not like the task in questions 6 and 7. By the way, note how you write the $sim$ in the notation $simeq$; it's backwards.
$endgroup$
– KCd
2 days ago
1
$begingroup$
Part of the official course description for your course says "Another goal is to make the students immerse in communicating mathematical thoughts (proofs, examples, counterexamples) in a written form." The way you answered questions 6 and 7 is not the standard way of describing isomorphisms at the level of your course (there are different notions of what it means to describe an isomorphism, depending on the assumed background of the audience). Since writing math properly is one of the goals of the course, I think the instructor was justified in taking off points in questions 6 & 7.
$endgroup$
– KCd
2 days ago
$begingroup$
As long as ya'll think it was appropriate of him, I won't say anything to him. I totally agree in that writing G$cong$H is not the same as writing that a specific function is an isomorphism in general. I guess I am assuming that because these groups are cyclic, have the same order, and have different generators, that this seems equivalent to stating what their maps are. I say this because writing <1> $cong$ <2> means 1 (in the first instance) must map to 2(in the second instance). And subsequently, the map is 1->2, 2->4, 3->1, 4->3, 0->0.
$endgroup$
– ness
2 days ago
$begingroup$
Also, I totally agree with you that he had a mark as well. I guess it just seemed odd that he didn't name a map from even to odd permutations but had less points taken off. Something I also notice is that he didn't justify #s_n=n!. Either way, I'm super stoked to hear that ya'll don't think this is a gender thing.
$endgroup$
– ness
2 days ago
|
show 1 more comment
$begingroup$
I just got a hw back from teacher and am curious about a marked mistake.
problem (6) was to "describe all isomorphisms from $Z_5$ to $Z_5$" and to (7) "describe all isomorphisms from $Z_4$ to $Z_4$."
I wrote that $Z_5 cong$ <$bar1$> $cong$ <$bar2$> $cong$ <$bar3$> $cong$ <$bar4$> where $bar a$ = a mod 5 and "<" , ">" denote 'the group generated by'.
Similarly, I wrote that $Z_4 cong$ <$hat1$> $cong$ <$hat3$> where $hat b$ = b mod 4 and "<" , ">" denote 'the group generated by'.
Teacher gave me half points for both problems because I 'did not explicitly write the map.' Isn't writing down that two cyclic groups with different generators are isomorphic a pretty explicit map? Does it seem a little overkill?
I don't know, I realize this isn't exactly the place for the question I'm about to ask, but I am just pretty floored and don't know of another community that would be more appropriate. Does this seem like a gender thing to you? My partner, an older male, has a fairly good understanding of algebra, but is receiving considerably higher marks. (He did problems 1 -3, while I did 4-7) -- see attached. I want to go to someone at the school, because it seems pretty biased to me, but would prefer to hear all of your opinions first.
http://math.sfsu.edu/gubeladze/classes/spring2019/335/homework%204%20(2019).pdf
https://imgur.com/VzItRZ7
https://imgur.com/zYNcpay
abstract-algebra group-theory cyclic-groups
asked Mar 9 at 2:04
nessness
426
$endgroup$
I just got a hw back from teacher and am curious about a marked mistake.
problem (6) was to "describe all isomorphisms from $Z_5$ to $Z_5$" and to (7) "describe all isomorphisms from $Z_4$ to $Z_4$."
I wrote that $Z_5 cong$ <$bar1$> $cong$ <$bar2$> $cong$ <$bar3$> $cong$ <$bar4$> where $bar a$ = a mod 5 and "<" , ">" denote 'the group generated by'.
Similarly, I wrote that $Z_4 cong$ <$hat1$> $cong$ <$hat3$> where $hat b$ = b mod 4 and "<" , ">" denote 'the group generated by'.
Teacher gave me half points for both problems because I 'did not explicitly write the map.' Isn't writing down that two cyclic groups with different generators are isomorphic a pretty explicit map? Does it seem a little overkill?
I don't know, I realize this isn't exactly the place for the question I'm about to ask, but I am just pretty floored and don't know of another community that would be more appropriate. Does this seem like a gender thing to you? My partner, an older male, has a fairly good understanding of algebra, but is receiving considerably higher marks. (He did problems 1 -3, while I did 4-7) -- see attached. I want to go to someone at the school, because it seems pretty biased to me, but would prefer to hear all of your opinions first.
http://math.sfsu.edu/gubeladze/classes/spring2019/335/homework%204%20(2019).pdf
https://imgur.com/VzItRZ7
https://imgur.com/zYNcpay
abstract-algebra group-theory cyclic-groups
abstract-algebra group-theory cyclic-groups
asked Mar 9 at 2:04
nessness
426
asked Mar 9 at 2:04
nessness
426
edited 2 days ago
ness
asked Mar 9 at 2:04
nessness
426
asked Mar 9 at 2:04
nessness
426
asked Mar 9 at 2:04
nessness
426
426
1
$begingroup$
The relation of being isomorphic means an isomorphism exists. That is not as specific as actually writing down an isomorphism. The point is that groups that are isomorphic have multiple isomorphisms between them (unless the groups have order 1 or 2). As a matter of notation, writing $G cong H$ is not the same as writing that a specific function $f colon G rightarrow H$ is an isomorphism. If someone learning English writes "I yesterday to store went" do you think that deserves close to full marks because it is clear what the intended meaning was despite mistakes?
$endgroup$
– KCd
2 days ago
1
$begingroup$
Now I looked at the links (I am a bit surprised you included your name at the top of the first page). I don't see a case of gender bias in the grades on these answers. You made the same error in questions 6 and 7, and got graded the same way for both of them. Your hmwk partner got 7/10 on question 1. In question 5, which you did and for which you got full marks, the task was to decide if the groups are isomorphic, not what an example of an isomorphism is. That is not like the task in questions 6 and 7. By the way, note how you write the $sim$ in the notation $simeq$; it's backwards.
$endgroup$
– KCd
2 days ago
1
$begingroup$
Part of the official course description for your course says "Another goal is to make the students immerse in communicating mathematical thoughts (proofs, examples, counterexamples) in a written form." The way you answered questions 6 and 7 is not the standard way of describing isomorphisms at the level of your course (there are different notions of what it means to describe an isomorphism, depending on the assumed background of the audience). Since writing math properly is one of the goals of the course, I think the instructor was justified in taking off points in questions 6 & 7.
$endgroup$
– KCd
2 days ago
$begingroup$
As long as ya'll think it was appropriate of him, I won't say anything to him. I totally agree in that writing G$cong$H is not the same as writing that a specific function is an isomorphism in general. I guess I am assuming that because these groups are cyclic, have the same order, and have different generators, that this seems equivalent to stating what their maps are. I say this because writing <1> $cong$ <2> means 1 (in the first instance) must map to 2(in the second instance). And subsequently, the map is 1->2, 2->4, 3->1, 4->3, 0->0.
$endgroup$
– ness
2 days ago
$begingroup$
Also, I totally agree with you that he had a mark as well. I guess it just seemed odd that he didn't name a map from even to odd permutations but had less points taken off. Something I also notice is that he didn't justify #s_n=n!. Either way, I'm super stoked to hear that ya'll don't think this is a gender thing.
$endgroup$
– ness
2 days ago
|
show 1 more comment
1
$begingroup$
The relation of being isomorphic means an isomorphism exists. That is not as specific as actually writing down an isomorphism. The point is that groups that are isomorphic have multiple isomorphisms between them (unless the groups have order 1 or 2). As a matter of notation, writing $G cong H$ is not the same as writing that a specific function $f colon G rightarrow H$ is an isomorphism. If someone learning English writes "I yesterday to store went" do you think that deserves close to full marks because it is clear what the intended meaning was despite mistakes?
$endgroup$
– KCd
2 days ago
1
$begingroup$
Now I looked at the links (I am a bit surprised you included your name at the top of the first page). I don't see a case of gender bias in the grades on these answers. You made the same error in questions 6 and 7, and got graded the same way for both of them. Your hmwk partner got 7/10 on question 1. In question 5, which you did and for which you got full marks, the task was to decide if the groups are isomorphic, not what an example of an isomorphism is. That is not like the task in questions 6 and 7. By the way, note how you write the $sim$ in the notation $simeq$; it's backwards.
$endgroup$
– KCd
2 days ago
1
$begingroup$
Part of the official course description for your course says "Another goal is to make the students immerse in communicating mathematical thoughts (proofs, examples, counterexamples) in a written form." The way you answered questions 6 and 7 is not the standard way of describing isomorphisms at the level of your course (there are different notions of what it means to describe an isomorphism, depending on the assumed background of the audience). Since writing math properly is one of the goals of the course, I think the instructor was justified in taking off points in questions 6 & 7.
$endgroup$
– KCd
2 days ago
$begingroup$
As long as ya'll think it was appropriate of him, I won't say anything to him. I totally agree in that writing G$cong$H is not the same as writing that a specific function is an isomorphism in general. I guess I am assuming that because these groups are cyclic, have the same order, and have different generators, that this seems equivalent to stating what their maps are. I say this because writing <1> $cong$ <2> means 1 (in the first instance) must map to 2(in the second instance). And subsequently, the map is 1->2, 2->4, 3->1, 4->3, 0->0.
$endgroup$
– ness
2 days ago
$begingroup$
Also, I totally agree with you that he had a mark as well. I guess it just seemed odd that he didn't name a map from even to odd permutations but had less points taken off. Something I also notice is that he didn't justify #s_n=n!. Either way, I'm super stoked to hear that ya'll don't think this is a gender thing.
$endgroup$
– ness
2 days ago
1
1
$begingroup$
The relation of being isomorphic means an isomorphism exists. That is not as specific as actually writing down an isomorphism. The point is that groups that are isomorphic have multiple isomorphisms between them (unless the groups have order 1 or 2). As a matter of notation, writing $G cong H$ is not the same as writing that a specific function $f colon G rightarrow H$ is an isomorphism. If someone learning English writes "I yesterday to store went" do you think that deserves close to full marks because it is clear what the intended meaning was despite mistakes?
$endgroup$
– KCd
2 days ago
$begingroup$
The relation of being isomorphic means an isomorphism exists. That is not as specific as actually writing down an isomorphism. The point is that groups that are isomorphic have multiple isomorphisms between them (unless the groups have order 1 or 2). As a matter of notation, writing $G cong H$ is not the same as writing that a specific function $f colon G rightarrow H$ is an isomorphism. If someone learning English writes "I yesterday to store went" do you think that deserves close to full marks because it is clear what the intended meaning was despite mistakes?
$endgroup$
– KCd
2 days ago
1
1
$begingroup$
Now I looked at the links (I am a bit surprised you included your name at the top of the first page). I don't see a case of gender bias in the grades on these answers. You made the same error in questions 6 and 7, and got graded the same way for both of them. Your hmwk partner got 7/10 on question 1. In question 5, which you did and for which you got full marks, the task was to decide if the groups are isomorphic, not what an example of an isomorphism is. That is not like the task in questions 6 and 7. By the way, note how you write the $sim$ in the notation $simeq$; it's backwards.
$endgroup$
– KCd
2 days ago
$begingroup$
Now I looked at the links (I am a bit surprised you included your name at the top of the first page). I don't see a case of gender bias in the grades on these answers. You made the same error in questions 6 and 7, and got graded the same way for both of them. Your hmwk partner got 7/10 on question 1. In question 5, which you did and for which you got full marks, the task was to decide if the groups are isomorphic, not what an example of an isomorphism is. That is not like the task in questions 6 and 7. By the way, note how you write the $sim$ in the notation $simeq$; it's backwards.
$endgroup$
– KCd
2 days ago
1
1
$begingroup$
Part of the official course description for your course says "Another goal is to make the students immerse in communicating mathematical thoughts (proofs, examples, counterexamples) in a written form." The way you answered questions 6 and 7 is not the standard way of describing isomorphisms at the level of your course (there are different notions of what it means to describe an isomorphism, depending on the assumed background of the audience). Since writing math properly is one of the goals of the course, I think the instructor was justified in taking off points in questions 6 & 7.
$endgroup$
– KCd
2 days ago
$begingroup$
Part of the official course description for your course says "Another goal is to make the students immerse in communicating mathematical thoughts (proofs, examples, counterexamples) in a written form." The way you answered questions 6 and 7 is not the standard way of describing isomorphisms at the level of your course (there are different notions of what it means to describe an isomorphism, depending on the assumed background of the audience). Since writing math properly is one of the goals of the course, I think the instructor was justified in taking off points in questions 6 & 7.
$endgroup$
– KCd
2 days ago
$begingroup$
As long as ya'll think it was appropriate of him, I won't say anything to him. I totally agree in that writing G$cong$H is not the same as writing that a specific function is an isomorphism in general. I guess I am assuming that because these groups are cyclic, have the same order, and have different generators, that this seems equivalent to stating what their maps are. I say this because writing <1> $cong$ <2> means 1 (in the first instance) must map to 2(in the second instance). And subsequently, the map is 1->2, 2->4, 3->1, 4->3, 0->0.
$endgroup$
– ness
2 days ago
$begingroup$
As long as ya'll think it was appropriate of him, I won't say anything to him. I totally agree in that writing G$cong$H is not the same as writing that a specific function is an isomorphism in general. I guess I am assuming that because these groups are cyclic, have the same order, and have different generators, that this seems equivalent to stating what their maps are. I say this because writing <1> $cong$ <2> means 1 (in the first instance) must map to 2(in the second instance). And subsequently, the map is 1->2, 2->4, 3->1, 4->3, 0->0.
$endgroup$
– ness
2 days ago
$begingroup$
Also, I totally agree with you that he had a mark as well. I guess it just seemed odd that he didn't name a map from even to odd permutations but had less points taken off. Something I also notice is that he didn't justify #s_n=n!. Either way, I'm super stoked to hear that ya'll don't think this is a gender thing.
$endgroup$
– ness
2 days ago
$begingroup$
Also, I totally agree with you that he had a mark as well. I guess it just seemed odd that he didn't name a map from even to odd permutations but had less points taken off. Something I also notice is that he didn't justify #s_n=n!. Either way, I'm super stoked to hear that ya'll don't think this is a gender thing.
$endgroup$
– ness
2 days ago
|
show 1 more comment
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1
$begingroup$
The relation of being isomorphic means an isomorphism exists. That is not as specific as actually writing down an isomorphism. The point is that groups that are isomorphic have multiple isomorphisms between them (unless the groups have order 1 or 2). As a matter of notation, writing $G cong H$ is not the same as writing that a specific function $f colon G rightarrow H$ is an isomorphism. If someone learning English writes "I yesterday to store went" do you think that deserves close to full marks because it is clear what the intended meaning was despite mistakes?
$endgroup$
– KCd
2 days ago
1
$begingroup$
Now I looked at the links (I am a bit surprised you included your name at the top of the first page). I don't see a case of gender bias in the grades on these answers. You made the same error in questions 6 and 7, and got graded the same way for both of them. Your hmwk partner got 7/10 on question 1. In question 5, which you did and for which you got full marks, the task was to decide if the groups are isomorphic, not what an example of an isomorphism is. That is not like the task in questions 6 and 7. By the way, note how you write the $sim$ in the notation $simeq$; it's backwards.
$endgroup$
– KCd
2 days ago
1
$begingroup$
Part of the official course description for your course says "Another goal is to make the students immerse in communicating mathematical thoughts (proofs, examples, counterexamples) in a written form." The way you answered questions 6 and 7 is not the standard way of describing isomorphisms at the level of your course (there are different notions of what it means to describe an isomorphism, depending on the assumed background of the audience). Since writing math properly is one of the goals of the course, I think the instructor was justified in taking off points in questions 6 & 7.
$endgroup$
– KCd
2 days ago
$begingroup$
As long as ya'll think it was appropriate of him, I won't say anything to him. I totally agree in that writing G$cong$H is not the same as writing that a specific function is an isomorphism in general. I guess I am assuming that because these groups are cyclic, have the same order, and have different generators, that this seems equivalent to stating what their maps are. I say this because writing <1> $cong$ <2> means 1 (in the first instance) must map to 2(in the second instance). And subsequently, the map is 1->2, 2->4, 3->1, 4->3, 0->0.
$endgroup$
– ness
2 days ago
$begingroup$
Also, I totally agree with you that he had a mark as well. I guess it just seemed odd that he didn't name a map from even to odd permutations but had less points taken off. Something I also notice is that he didn't justify #s_n=n!. Either way, I'm super stoked to hear that ya'll don't think this is a gender thing.
$endgroup$
– ness
2 days ago