# Appendix: Extra Topics

$$N_G(H) / C_G(H)$$ 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 $$R_i \otimes_k R_{n-j} \to R_n$$.