How Can I Differentiate Various Group Types in Algebra?

[Charles007]

THese days. I am trying to tidy up with my summary notes.

I got surjective homomophism, isomorphic, and cyclic group, Subgroup, normal group, symmetric group, Quaternion group, and quotient group.

How to distinguish them. Is there any tric to understand and remember these. I am completely mess up with these things. what I should do...!~~ I being revising for 10days.

Any one give me some suggestion.

 

[Charles007]

One more question.

How, the supremum, the least upper bound.

Infimum, , supremum.

what is it mean?

Attempt.

for example,

1,. (0,15]

2, {n/2n+1|n belongs to nautral number (N)}

 

[VeeEight]

An isomorphism is a bijective homomorphism.
A subgroup is a group that is a subset of some 'larger' group (it is usually studied in relation to the original group)
A group is normal is every left coset is a right coset (rearranging the formula is useful for solving common group theory problems). Quotient groups are related to normal groups and deal with cosets.
The quaternion group is a nonabelian group with interesting properties
The symmetric group is a group of permutations.

The supremum of a set is the least upper bound of the set. That is, out of all the upper bounds of your set, the supremum is the 'smallest'.
The infimum is the opposite and is the greatest lower bound of the set.
for (0, 15], the supremum is 15 and infimum is 0

 

[Charles007]

An isomorphism is a bijective homomorphism.
A subgroup is a group that is a subset of some 'larger' group (it is usually studied in relation to the original group)
A group is normal is every left coset is a right coset (rearranging the formula is useful for solving common group theory problems). Quotient groups are related to normal groups and deal with cosets.
The quaternion group is a nonabelian group with interesting properties
The symmetric group is a group of permutations.

The supremum of a set is the least upper bound of the set. That is, out of all the upper bounds of your set, the supremum is the 'smallest'.
The infimum is the opposite and is the greatest lower bound of the set.
for (0, 15], the supremum is 15 and infimum is 0

Thank you very much. Now I know what I should do next.

Collect all defination of group and connect them using your collection.

One more question. suppose we have a set A (-12,3]. (-x)------(-12)---------0-------------3----------------------------x

They minum for set A is -12, and infimum is -12. -12 is not closed, and 3 is closed.

The supremum of a set is the least upper bound of the set. So the 3 is supremum, the least upper bound [3,x) . also 3 is the max.

Am I right?

 

[Charles007]

Let A and B be non-empty and bounded subsets of R.

What it is mean by bounded?

 

[matticus]

They minum for set A is -12, and infimum is -12. -12 is not closed, and 3 is closed.

Am I right?

a number cannot be closed. sets are closed if they contain all their limit points. the set you list is not closed, since 12 is a limit point and 12 is not in the set.

 

[VeeEight]

One more question. suppose we have a set A (-12,3]. (-x)------(-12)---------0-------------3----------------------------x

They minum for set A is -12, and infimum is -12. -12 is not closed, and 3 is closed.

The supremum of a set is the least upper bound of the set. So the 3 is supremum, the least upper bound [3,x) . also 3 is the max.

Am I right?

Yes 3 is the supremum and -12 is the infimum. The set has no minimum.

Let A and B be non-empty and bounded subsets of R.

What it is mean by bounded?

Set A is bounded above if there is some x in R such that a < x for all a in A. Bounded below is defined similarly and a set is bounded if it is bounded above and below.