Basic Examples (5) 

Embed CyclicGroup[2] in CyclicGroup[4] via (12)↦(13)(24):

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Click on + to reveal more information about the homomorphism. For example, ϕ is injective but not surjective. Obtain the underlying mapping:

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Retrieve the domain or co-domain of a homomorphism:

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Find the kernel of this homomorphism, indeed the kernel is singleton thus ϕ is an embedding:

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This is the trivial group:

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Find the image group:

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Try to specify an invalid homomorphism:

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The package efficiently works with large groups. For example, here is the identity homomorphism from SymmetricGroup[10] to itself:

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Its image and kernel groups:

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Scope (2) 

Specify the homomorphism using an association:

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Build the homomorphism:

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This is the identity homomorphism (an isomorphism, since it is both injective and surjective):

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Specify an endomorphism (from G to itself):

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Applications (3) 

Construct the sign homomorphism from SymmetricGroup[4] to CyclicGroup[2]:

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Odd generators must map to (12):

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Find all even permutations:

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Confirm that there are exactly 12 even permutations:

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Quotient DihedralGroup[3] by CyclicGroup[2] sending all rotations to ε and reflections to (12):

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Again, kernel is all 3 rotations:

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For an AbelianGroup to itself, the map x↦x^k is a homomorphism:

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Its kernel:

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Properties and Relations (4) 

GroupHomomorphism preserves operation:

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GroupHomomorphism preserves inverses:

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The specified images of generators must belong to G₂:

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Applying the homomorphism to an arbitrary object returns unevaluated:

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Replace x by an element:

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Possible Issues (3) 

Consider the homomorphism from SymmetricGroup[3] to CyclicGroup[6]:

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We mapped an order 2 generator to order 2 element, and 3 to 3. This violates the group relations of SymmetricGroup[3]. Consider for example:

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But:

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