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Preimage of a Subgroup is a Subgroup (Use One Step Subgroup Test)

Given a group isomorphism (or homomorphism) φ: G → G̅, if K̅ is a subgroup of G̅, then the preimage (a.k.a. inverse image) φ^-1(K̅) is a subgroup of G. We prove this with the one-step subgroup test. This works for homomorphisms as well. There is no need to assume φ is one-to-one or onto, though the inverse function φ^-1 does not exist in that case. We can still use the notation φ^-1(K̅), however. Links and resources =============================== 🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinn... 🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/ 🔴 Follow me on Twitter:   / billkinneymath   🔴 Follow me on Instagram:   / billkinneymath   🔴 You can support me by buying "Infinite Powers, How Calculus Reveals the Secrets of the Universe", by Steven Strogatz, or anything else you want to buy, starting from this link: https://amzn.to/3eXEmuA. 🔴 Check out my artist son Tyler Kinney's website: https://www.tylertkinney.co/ 🔴 Desiring God website: https://www.desiringgod.org/ AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.