\(N_G(H) / C_G(H)\) is isomorphic to a subgroup of \(\mathop{\mathrm{Aut}}(H)\).
If for every proper \(H < G\), \(H{~\trianglelefteq~}N_G(H)\) is again proper, then “normalizers grow” in \(G\).
Characteristic Subgroups
Normality is not transitive!
I.e. if \(H{~\trianglelefteq~}G\) and \(N{~\trianglelefteq~}H\), it’s not necessarily the case that \(N{~\trianglelefteq~}G\).
A subgroup \(H\leq G\) is characteristic in \(G\), written \(H \operatorname{ch}G\), iff for every \(\phi \in \mathop{\mathrm{Aut}}(G)\), \(\phi(H) \leq H\). Equivalently, \(\phi(H) = H\). I.e. \(H\) is fixed (not necessarily pointwise) under every automorphism of the ambient group \(G\).
Characteristic subgroups are normal, because \(\psi_g({-}) \coloneqq g({-})g^{-1}\) is an (inner) automorphic of \(G\). Not every normal subgroup is characteristic: take \(G \coloneqq H_1 \times H_2\) and \(\psi(x, y) = (y, x)\).
Characteristic subgroups of normal subgroups are normal, i.e. if \(H{~\trianglelefteq~}G\) and \(N \operatorname{ch}H\), then \(N{~\trianglelefteq~}G\).
\(A \operatorname{ch}B {~\trianglelefteq~}C \implies A{~\trianglelefteq~}C\):
- \(A\operatorname{ch}B\) iff \(A\) is fixed by every \(\psi\in \mathop{\mathrm{Aut}}(B)\)., WTS \(cAc^{-1}= A\) for all \(c\in C\).
- Since \(B{~\trianglelefteq~}C\), the automorphism \(\psi({-}) \coloneqq c({-})c^{-1}\) descends to an element of \(\mathop{\mathrm{Aut}}(B)\).
- Then \(\psi(A) = A\) since \(A\operatorname{ch}B\), so \(cAc^{-1}= A\) and \(A{~\trianglelefteq~}C\).
For any group \(G\), \begin{align*} Z(G) \operatorname{ch}G .\end{align*}
Let \(\psi \in \mathop{\mathrm{Aut}}(H)\) and \(x=\psi(y)\in \psi(Z(H))\) so \(y\in Z(H)\), then for arbitrary \(h\in H\), \begin{align*} \psi(y)h &= \psi(y) (\psi \circ \psi^{-1})(h) \\ &= \psi( y \cdot \psi^{-1}(h) ) \\ &= \psi( \psi^{-1}(h) \cdot y ) && \text{since } \psi^{-1}(h)\in H, \, y\in Z(H) \\ &= h\psi(y) .\end{align*}
Normal Closures and Cores
The smallest normal subgroup of \(G\) containing \(H\): \begin{align*} H^G \coloneqq\{gHg^{-1}: g\in G\} = \bigcap \left\{{N: H \leq N {~\trianglelefteq~}G }\right\} .\end{align*}
The largest normal subgroup of \(G\) containing \(H\): \begin{align*} H_G = \bigcap_{g\in G} gHg^{-1} = \left\langle{ N: N {~\trianglelefteq~}G ~\&~ N \leq H}\right\rangle = \ker \psi .\end{align*} where \begin{align*} \psi: G &\to \mathop{\mathrm{Aut}}(G/H) \\ g &\mapsto (xH\mapsto gxH) \end{align*}
If \(H{~\trianglelefteq~}G\) and \(P \in \mathrm{Syl}_p(G)\), then \(H N_G(P) = G\) and \([G: H]\) divides \({\left\lvert {N_G(P)} \right\rvert}\).
Exercises
Show that \(Z(G) \leq G\) is always characteristic.
Let \(\psi\in \mathop{\mathrm{Aut}}(G)\). For one containment, we can show \(\psi(g) = h = h\psi(g)\) for all \(\psi(g) \in \psi(G)\) and \(h\in G\). This is a computation: \begin{align*} \psi(g) h &= \psi(g) (\psi \psi^{-1})(h) \\ &= \psi( g ) \psi( \psi ^{-1}(h)) \\ &= \psi( \psi^{-1}(h) g) \\ &= (\psi\psi^{-1})(h) \psi(g) \\ &= h\psi(g) .\end{align*} This yields \(\psi(Z(G)) \subseteq Z(G)\). Applying the same argument to \(\psi^{-1}\) yields \(\psi^{-1}(Z(G)) \subseteq Z(G)\). Since \(\psi\) is a bijection, \(\psi\psi^{-1}(A) = A\) for all \(A\leq G\), so \(Z(G) \subseteq \psi(Z(G))\).
Nilpotent Groups
A group \(G\) is nilpotent iff \(G\) has a terminating upper central series.
Moral: the adjoint map is nilpotent.
\(G\) is nilpotent iff \(G\) has an upper central series terminating at \(G\).
\(G\) is nilpotent iff \(G\) has a lower central series terminating at \(1\).
A group \(G\) is nilpotent iff all of its Sylow \(p{\hbox{-}}\)subgroups are normal for every \(p\) dividing \({\left\lvert {G} \right\rvert}\).
A group \(G\) is nilpotent iff every maximal subgroup is normal.
For \(G\) a finite group, TFAE:
- \(G\) is nilpotent
- Normalizers grow, i.e. if \(H < G\) is proper then \(H < N_G(H)\).
- Every Sylow-p subgroup is normal
- \(G\) is the direct product of its Sylow p-subgroups
- Every maximal subgroup is normal
- \(G\) has a terminating Lower Central Series
- \(G\) has a terminating Upper Central Series
-
Nilpotent groups satisfy the 2 out of 3 property.
\todo{Todo. Specify.} - \(G\) has normal subgroups of order \(d\) for every \(d\) dividing \({\left\lvert {G} \right\rvert}\)
Rings
A commutative Noetherian ring \(R\) is Gorenstein iff \(R\) viewed as an \(R{\hbox{-}}\)module has finite injective dimension.
If \(R\in {\mathsf{gr}\,} {}_{k} \mathsf{Alg}\) with \(\dim_k R < \infty\), then \(R\) decomposes as \(R = R_0 \oplus R_1 \oplus \cdots R_n\) with \(R_0 \coloneqq k\), and \(R\) is Gorenstein iff \(R\) satisfies “Poincaré duality”: \(\dim_k R_0 = \dim_k R_m = 1\) and there is a perfect pairing \(R_i \otimes_k R_{n-j} \to R_n\).