Dummit+and+foote+solutions+chapter+4+overleaf+full __hot__ 【2024】

\beginproof Transitive: For any $aH, bH$, $(ba^-1)\cdot aH = bH$. Kernel: $g\in \ker \iff gxH = xH \ \forall x \iff x^-1gx \in H \ \forall x \iff g \in \bigcap_x\in G xHx^-1$. \endproof

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\subsection*Exercise 20 State the class equation for a finite group $G$: \[ |G| = |Z(G)| + \sum [G : C_G(g_i)], \] where the sum runs over representatives of conjugacy classes of size $>1$. \beginproof Transitive: For any $aH, bH$, $(ba^-1)\cdot aH