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## Prove that a given function is a ring homomorphism and describe its kernel

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.7 Let $R \subseteq M_2(\mathbb{Z})$ be the subring of all upper triangular matrices. Prove that the map…

## Generalized coordinate subgroups of a direct product

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.2 Let $G_1,G_2,\ldots,G_n$ be groups and let $G = \times_{i=1}^k G_i$. Let $I$ be a proper nonempty…

## The center of a direct product is the direct product of the centers

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.1 Show that the center of a direct product is the direct product of the centers: Z(G_1…

## Compute some orbits of an action by Sym(4) on polynomials in four variables

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.6 As in Exercise 2.2.12, let $S_4$ act on the set $R$ of all polynomials with integer…