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Widget
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21 Nov 2014, 10:07 am

Could someone give some examples of the what this example means?

"Relations among abstract structures of mathematical operations (e.g., detecting structural isomorphisms between groups of mathematical operations in disparate structures of mathematical operations ."

this is from a developmental psychology textbook under
Table 6.1 Developmental Transformations of Hierarchical Complexity in Three Domains



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21 Nov 2014, 11:18 am

Holy smoke! Now I understand, why it takes good mathematical skills to get through psychology studies at university.


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21 Nov 2014, 11:29 am

Widget wrote:
Could someone give some examples of the what this example means?

"Relations among abstract structures of mathematical operations (e.g., detecting structural isomorphisms between groups of mathematical operations in disparate structures of mathematical operations ."

this is from a developmental psychology textbook under
Table 6.1 Developmental Transformations of Hierarchical Complexity in Three Domains


Group isomoprhism: A one to one onto mapping from group G to group G' that preserves group structure. Let h be such an isomorphism. then for x, y in G and where * is the group operator h(x*y) = h(x)*'h(y) where *' the group operator of G'.

h(x inverse) = h(x) inverse and h(identity element of G) is the identity element of G'

A homomorphism is like an isomorphism except h has to be a mapping from G onto G'. It need not be one to one

For more one Groups see any book on abstract algebra. Examples of groups: The real numbers where +is the operation and 0 is the identity element.



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24 Nov 2014, 6:56 pm

I think in English its saying that while before age 25 a person can perform quite complex calculations its only later in life they develop the necessary skills to "understand" a complex system with one look as the much more robust subconscious learning and deeply ingrained memories of previously performed routine calculations allow the analysis to be done practically instantly.

For example a young novice engineer might be able to do the math and determine that a bridge wont hold the required weight but only an older one with much more experience would be able to glance at the blueprint and immediately "know" it wont hold with the same degree of certainty.


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25 Nov 2014, 3:19 am

MorganFTL wrote:
I think in English its saying that while before age 25 a person can perform quite complex calculations its only later in life they develop the necessary skills to "understand" a complex system with one look as the much more robust subconscious learning and deeply ingrained memories of previously performed routine calculations allow the analysis to be done practically instantly.

For example a young novice engineer might be able to do the math and determine that a bridge wont hold the required weight but only an older one with much more experience would be able to glance at the blueprint and immediately "know" it wont hold with the same degree of certainty.


I'm sorry to say that what you write here does not have any bearing whatsoever on the original question.



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25 Nov 2014, 3:28 am

ruveyn wrote:
A homomorphism is like an isomorphism except h has to be a mapping from G onto G'. It need not be one to one.


I don't think a group homomorphism is required to be onto.



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25 Nov 2014, 8:14 pm

SweetTooth wrote:
ruveyn wrote:
A homomorphism is like an isomorphism except h has to be a mapping from G onto G'. It need not be one to one.


I don't think a group homomorphism is required to be onto.


Good point. An into mapping will do just fine. The image of the homomorphism has the structure of the group from which it is mapped or a structure -like- the group from which it is mapped.



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26 Nov 2014, 2:22 pm

Yes, indeed. In fact, h(G) is isomorphic to G/Ker(h).



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26 Nov 2014, 7:53 pm

SweetTooth wrote:
Yes, indeed. In fact, h(G) is isomorphic to G/Ker(h).
e f

The way to find all the homomorphs of a given group is to get all the normal subgroups and for each develop the quotient group. So all the homomorphs can be found by inside work. I think that is semi-remarkable.

It is analogous to learning the curvature of a space without ever leaving the space or embedding it in a larger space.

We never had to go into orbit to find out that the earth was a spheroid

ruveyn



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27 Nov 2014, 6:04 am

ruveyn wrote:
SweetTooth wrote:
Yes, indeed. In fact, h(G) is isomorphic to G/Ker(h).
e f

The way to find all the homomorphs of a given group is to get all the normal subgroups and for each develop the quotient group. So all the homomorphs can be found by inside work. I think that is semi-remarkable.

It is analogous to learning the curvature of a space without ever leaving the space or embedding it in a larger space.

We never had to go into orbit to find out that the earth was a spheroid

ruveyn


Sorry, but what does "e f" mean?

That is not semi-remarkable, it is remarkable. I find results that characterise apparantly extrinsic qualities in an intrinsic manner among the most beautiful in mathematics.