List questions from abelian-groups

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Prove that set $A$ a finite abelian group and $B$ multiplicative group of all homomorphisms from $A$ to units of $C$ are isomorphic

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$p$-primary part of an abelian group

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Find generators of finite abelian group, s.t. subgroup embeds diagonally

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Prove that for homomorphism $\phi:G\to H$ if $G_{1}\leq G$ is abelian, then $\phi(G_{1})$ is abelian.

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Order of element in a group

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Prove that the only automorphism of order 2 of $\mathbb{Z}_q$ is $m \mapsto -m$ for $q = 3$ and $q = 5$ and $q = 7$

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Books Recommendation for Special Group Theory Topics

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Proof verification: In finite abelian groups, any subgroup generated by an element of maximal order is a direct summand.

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Relationship of size of quotient group to matrix determinant

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List Question

Bases of Subgroups of Infinitely Generated Free Abelian Groups

Fundamental Theorem of Abelian Groups - intuition regarding Lemma

Prove that set $A$ a finite abelian group and $B$ multiplicative group of all homomorphisms from $A$ to units of $C$ are isomorphic

$p$-primary part of an abelian group

Is the specific group abelian?

Find generators of finite abelian group, s.t. subgroup embeds diagonally

Prove that for homomorphism $\phi:G\to H$ if $G_{1}\leq G$ is abelian, then $\phi(G_{1})$ is abelian.

Group of order $4$ such that $x^2=1$ for all $x$?

Order of element in a group

Free group generated by two generators is isomorphic to product of two infinite cyclic groups

Torsion in weird groups

Prove that the only automorphism of order 2 of $\mathbb{Z}_q$ is $m \mapsto -m$ for $q = 3$ and $q = 5$ and $q = 7$

Books Recommendation for Special Group Theory Topics

Proof verification: In finite abelian groups, any subgroup generated by an element of maximal order is a direct summand.

Relationship of size of quotient group to matrix determinant

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