Subgroups and cosets
\(G\): a group.
\(H \leq G\): a subgroup of \(G\).
\(gH = \{gh \mid h \in H \}\): left coset of \(H\) with respect to \(g \in G\).
\(Hg = \{hg \mid h \in H \}\): right coset of \(H\) with respect to \(g \in G\).
\(L = \{gH \mid g \in G \}\): set of all left cosets of \(H\).
\(R = \{Hg \mid g \in G \}\): set of all right cosets of \(H\).
Normal subgroup
\(N\): a subgroup of \(G\).
\(gN = Ng \quad \forall \, g \in G\): \(N\) is a normal subgroup of \(G\).
\(N \trianglelefteq G\): notation for normality.
\(G \trianglelefteq G\): normal subgroup of itself.
\(\{ 1 \} \trianglelefteq G\): trivial subgroup is normal.
e.g. Not normal
\(S_3 = \{ (), (1 \ 2), (1 \ 3), (2 \ 3), (1 \ 2 \ 3), (1 \ 3 \ 2) \}\): symmetric group.
\(S_2 = \{ (), (1 \ 2) \}\): symmetric group.
\(S_2 \leq S_3\): subgroup.
\((2 \ 3 ) S_2 = \{ (2 \ 3), (1 \ 2 \ 3) \}\): left coset.
\(S_2 (2 \ 3 ) = \{ (2 \ 3), (1 \ 3 \ 2) \}\): right coset.
\(S_2\) is not normal in \(S_3\).
e.g. Normal
\(S_3 = \{ (), (1 \ 2), (1 \ 3), (2 \ 3), (1 \ 2 \ 3), (1 \ 3 \ 2) \}\): symmetric group.
\(A_3 = \{ (), (1 \ 2 \ 3), (1 \ 3 \ 2) \}\): alternating group.
\(A_3 \leq S_3\): subgroup.
\((2 \ 3 ) A_3 = \{ (2 \ 3), (1 \ 2), (1 \ 3) \}\): left coset.
\(A_3 (2 \ 3 ) = \{ (2 \ 3), (1 \ 3), (1 \ 2) \}\): right coset.
\(A_3\) is normal in \(S_3\) but must check the others.