Questions about cosets etc. I don't understand my notes that my teacher gave me, so... please answer those questions in bold.
From my teacher's notes: "Suppose $H$ is a subgroup of $G$. For any $a\in G$, we define its associated left coset and right coset to be $$aH = \{ab | b \in H\}$$ $$Ha = \{ba | b\in H\}$$ respectively."
What is a coset?
What are the differences between left and right?
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Also from the notes: "Suppose $H$ is a subgroup of $G$. The left cosets $\{aH | a\in G\}$ define a partition of $G$, so are the right cosets."
What does "define a partition of $G$" mean?
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Also: "A function $f:G\rightarrow H$ where $G$, $H$ are groups is called a group homomorphism if $$f(a\cdot b)=f(a)\cdot f(b)$$ for all $a,b \in G$. It's called an isomorphism if it's bijective."
Can anyone explain the notation of "$f:G\rightarrow H$"?
What does bijective mean?
 A: The definitions of left / right coset are right there, and the different definitions make it clear that left and right coset are in general different.
A partition of a set $X$ is a collection of (non-empty) subsets of $X$ such that these subsets are pairwise disjoint while the union of all these subsets is $X$.
If $X,Y$ are sets, the notation $f\colon X\to Y$ expresses that $f$ is a function from $X$ to $Y$; that is, for each $x\in X$, $f$ defines some $f(x)\in Y$.
A function can be injective (or one-to-one) meaning that $f(x_1)=f(x_2)$ holds only when $x_1=x_2$; it can be surjective (or onto) if for every $y\in Y$ there is at least one $x\in X$ with $f(x)=y$; and if a function is both injective and surjective, it is called bijective.
A: In my opinion, Wikipedia is a good enough source to read & understand these basic concepts in Algebra.
(Cosets) https://en.wikipedia.org/wiki/Coset
(Partitions of a group) https://groupprops.subwiki.org/wiki/Left_cosets_partition_a_group
(Group Isomorphism)https://en.wikipedia.org/wiki/Group_isomorphism
