\alpha:G \to H is termed a surjective homomorphism of groups from G to H ... # \alpha is a defining ingredient::homomorphism of groups from G to ... ...
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===As the category of Lazard Lie rings===
The Lazard Lie category can ... Viewed this way, the category is a full subcategory of the category ... ...
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===As the category of Lazard Lie rings===
The Lazard Lie category can ... Viewed this way, the category is a full subcategory of the category ... ...
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#redirect equivalence of definitions of homomorphism of groups ... ...
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===In terms of a homomorphism of linear representations=== An equivalence of linear representations between a linear representation ... ...
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===As the category of Baer Lie rings===
The Baer Lie category can be ... Viewed this way, the category is a full subcategory of the category ... ...
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A homomorphism of groups is termed a monomorphism or an injective ... # It is injective as a map of sets
# Its Defining ingredient::kernel ... ...
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===As the category of LCS-Baer Lie rings===
The LCS-Baer Lie category ... Viewed this way, the category is a full subcategory of the category ... ...
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The category of groups forms a full subcategory of the category of ... # uses::Equivalence of definitions of homomorphism of groups ... ...
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The following are equivalent for an automorphism \sigma of a group G: # \sigma is a pushforwardable automorphism of G: For any homomorphism ... ...
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Homomorphism redirect here. For the more general definition of homomorphism ... Equivalence of definitions of homomorphism of groups
The textbook definition ... ...
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==The definitions we have to prove as equivalent==
===Definition as ... A group action of a group G on a set S is a map \alpha:G \times S ... ...
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A subgroup of a group is termed a fully invariant direct factor if ... subgroup having no nontrivial homomorphism to its quotient group ... ...
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==The definitions that we have to prove are equivalent== ... Then a map \varphi: G \to H is termed a homomorphism of groups if \varphi satisfies ... ...
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i.e., there exists an fact about::isotopy of magmas between G and H. e., there exists an fact about::isomorphism of magmas between G and H. ... ...
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The following are equivalent for an automorphism \sigma of a group G: pullbackable automorphism: For any homomorphism \rho:H \to G, there is ... ...
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A subgroup H of a group G is termed a characteristic direct factor ... # G is a nilpotent group and H is a characteristic direct factor of ... ...
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A subgroup of a group occurs as the fact about::kernel of a group ... A subgroup N of a group G occurs as the kernel of a group homomorphism ... ...
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A normal subgroup of finite index in a group is a subgroup satisfying ... # It is the kernel of a homomorphism to a finite group
# It is the ... ...
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===Equivalent definitions in tabular format===
{| class="sortable ... ===Equivalence of definitions===
equivalence of definitions of group ... ...
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