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On Oct 14, you were asked to bring an induction proof to class, which we then went over in class. You were then asked to submit a revision by the next class period.

I will ask you to continue to submit induction proofs until you have demonstrated mastery of this skill. Use the following sequence of problems to practice on. If you have already submitted, the first, and got feedback on it, then submit the second. If you have already submitted the second, then please work on the third. Etc. I'll add more problems to this list as needed.

Initial Induction List

Problem (Closed Under Function Composition)

Suppose that $H$ is a set of permutations of $X$. Suppose also that if $\alpha,\beta\in H$, then we must have $\alpha\circ \beta\in H$. Use induction to prove that for every $n\in \mathbb{N}$, if we know $\sigma_1,\sigma_2,\ldots,\sigma_n\in H$, then we must have $$\sigma_1\circ\sigma_2\circ\cdots\circ\sigma_n\in H.$$

Problem

We know that the inverse of $ab$ is $b^{−1}a^{−1}$. Use induction to prove that for every $n\in \mathbb{N}$ if we know $a_1,a_2,\ldots,a_n\in G$, then the inverse of $a_1a_2\cdots a_n$ is $a_n^{-1}\cdots a_2^{-1}a_1^{-1}$.

Problem

Suppose that $H$ is a subset of a group $G$ that is closed under the operation, which means we know that if $a,b\in H$, then $ab\in H$. Prove that for every $n\in \mathbb{N}$, if we have $a_1,a_2,\ldots,a_n\in H$, then we must have $a_1a_2\cdots a_n\in H$.

Problem

Suppose that $G$ is a group. Prove that for every $n\in \mathbb{N}$, if we have $H_1,H_2,\ldots,H_n$ are subgroups of $G$, then the intersection $H_1\cap H_2\cap\cdots\cap H_n$ is a subgroup of $G$. You may assume that $A\cap B$ is a subgroup of $G$ when $A$ and $B$ are subgroups of $G$.

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