I’m trying to understand Group Theory: perhaps over-optimistically, because I only have high-school maths (UK A-level).
I have some comprehension issues with basic definitions.
Very briefly: I understand that a group is a structure comprising a set of elements and a binary operator, denoted (S,∗), where S is the set and ∗ the operator, for example (ℤ,+). [Binary in the sense that it operates on any two elements of the set.] And it must meet four conditions: closure, associative, identity, inverse (just short-hands, but I’m assuming they’re comprehensible to people who understand group theory). My comprehension issues are with closure and inverse (at least, I think that's where my issues are).
Also very briefly: I understand that a subgroup is a subset of S with the same operator as its parent group, and which is itself a group as defined.
Closure (if I understand correctly): “If a and b are members of the group, then a∗b must also be a member of the group.” In other words, the group must be closed under its operator. So as I understand it: the infinitely large group (ℤ,+) is closed, but (1,2,3,+) is not closed, and thus cannot be a group. And (1,2,3,x) likewise can’t be a group, because although 1x2=2 which is in-group, 2x3=6 which is not-in-group. Does this mean that all groups must comprise infinitely large sets? I mean: when the operation is x, a x 1 will always be in-group; when the operation is +, a + 0 will always be in-group; but for all other instances of a x b or a + b to be in-group, the group must surely be infinitely large? And that doesn’t appear to be the case: at least, it surely can’t be the case for subgroups??? I think I must be misunderstanding something here.
Inverse (if I understand correctly): “The inverse of every element in the group must also be in the group, where the inverse of element a is the element that combines with a to give the identity.” So for example if the group’s operation is multiplication (hence identity element is 1) and a is 5, then a-1 = 0.2 must also be an element of the group. Is this example correct? I’m not sure: it seems to imply that (ℤ,x) can’t be a group. Which doesn’t sound right. Again, I think I must be misunderstanding something.
I’m sure these are very dumb questions, but I’d really appreciate any help: thanks so much.
I have some comprehension issues with basic definitions.
Very briefly: I understand that a group is a structure comprising a set of elements and a binary operator, denoted (S,∗), where S is the set and ∗ the operator, for example (ℤ,+). [Binary in the sense that it operates on any two elements of the set.] And it must meet four conditions: closure, associative, identity, inverse (just short-hands, but I’m assuming they’re comprehensible to people who understand group theory). My comprehension issues are with closure and inverse (at least, I think that's where my issues are).
Also very briefly: I understand that a subgroup is a subset of S with the same operator as its parent group, and which is itself a group as defined.
Closure (if I understand correctly): “If a and b are members of the group, then a∗b must also be a member of the group.” In other words, the group must be closed under its operator. So as I understand it: the infinitely large group (ℤ,+) is closed, but (1,2,3,+) is not closed, and thus cannot be a group. And (1,2,3,x) likewise can’t be a group, because although 1x2=2 which is in-group, 2x3=6 which is not-in-group. Does this mean that all groups must comprise infinitely large sets? I mean: when the operation is x, a x 1 will always be in-group; when the operation is +, a + 0 will always be in-group; but for all other instances of a x b or a + b to be in-group, the group must surely be infinitely large? And that doesn’t appear to be the case: at least, it surely can’t be the case for subgroups??? I think I must be misunderstanding something here.
Inverse (if I understand correctly): “The inverse of every element in the group must also be in the group, where the inverse of element a is the element that combines with a to give the identity.” So for example if the group’s operation is multiplication (hence identity element is 1) and a is 5, then a-1 = 0.2 must also be an element of the group. Is this example correct? I’m not sure: it seems to imply that (ℤ,x) can’t be a group. Which doesn’t sound right. Again, I think I must be misunderstanding something.
I’m sure these are very dumb questions, but I’d really appreciate any help: thanks so much.
