There is something that I do not understand about the definition of a left coset.
Let $\,G\,$ be a group and $\,H\,$ be a subgroup of $\,G\,$. Then the subset $\,aH=\{ah |h \in H\}\subseteq G\,$ is the left coset of $\,H\,$ containing $\,a.$
What is "$\,a\,$" in this definition? What does it represent? Can anyone help?
Thanks.
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$\begingroup$The "$a$" in the definition is any element of $G$: .
So the left coset $\,aH\subseteq G\,$ is the set of all elements in the left coset $aH$, which for a given $\,a \in G\,$ and every element $h_i \in H$, is the set of all $ah_i$.
E.g. Take a small subgroup of $S_3$ : $\;H = \langle (12)\rangle = \{id, (12)\} \leq S_3.\,$ There are three left (respectively right) cosets of $\,H$ in $\,S_3$. One coset is $\,H\,$ itself. The other cosets are $\,(13)H = (123)H\,$ and $\,(23)H = (132)H$.
You'll see that for any subgroup $\,H \leq G$, every element of $\,G\,$ will belong to one and only one left (respectively right) coset of $\,H\,$ in $\,G.\,$ And the union of all left cosets of $H$ in $G$ (respectively the union of all right cosets of $H$ in $G$) is $G$. That is, the left (respectively right) cosets of $H$ in $G$ partition $G$.
You'll can find a nice definition of "coset" and some examples here, as well.
$\endgroup$ 4 $\begingroup$$a$ can be any element of $G$. For example, let's take the symmetries of the square, $D_4$, and a subgroup of it: $H = \{e, r_1, r_2, r_3\}$. We'll pick $f_v$ for our $g$. $gH = \{ gh : h \in H\} = \{ f_ve, f_vr_1, f_vr_2, f_vr_3 \} = \{ f_v, f_d, f_h, f_c \}$. $f_v$ turns the group of rotations into a set of flips (which is not a group).
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