Let G be simple, |G| is not 2, and ϕ a homomorphism from G to H. If H contains a normal subgroup A of index 2, then ϕ(G) ≤ A .
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Problem from "The Theory of Finite Groups" by Kurzweil and Stellmacher
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$|G|=24$ prove that $G$ is not simple
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$f: G→G$ defined by $f(x) =x^2$ is a homomorphism if and only if $G$ is abelian.
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Nonempty subset H of group G is subgroup iff $ab^{-1} \in H $ for any $a,b \in H$
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Normality of subgroups and Sylow groups under homomorphisms...
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Counterexamples for $G \times H$ isomorphic to $G$ implies $H$ trivial.
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the image of normal subgroups
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H is a subgroup of G and G' is a subgroup of H. Prove H is normal in G.
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Let $f : G → G_1$ be a surjective homomorphism (also called epimorphism) from $G$ to another group $G_1$. Prove that $f(Z(G)) \subseteq Z(G_1)$.
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If $H_1, H_2\leq G$ are such that $H_1\cong H_2$ then $G/H_1\cong G/H_2$?
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Normal subgroup of a normal subgroup
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If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
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Existence of finite indexed normal subgroup for a given finite indexed subgroup.
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A subgroup with an index of 2 is normal to the group.
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proof verification: If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian
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Normal Subgroup Counterexample
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If a maximal subgroup is normal, it has prime index