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# Kernel of Phi and Homomorphism

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Let phi is a homomorphism from Z30 onto a group order 3. Determine the kernel of phi. Find all generators of the kernel of phi.

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The solution determines the kernel of phi. Find all generators of the kernel of phi.

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Z30 = {0, 1, ... 29}, the group of integers mod 30, with the composition (+), defined as:
a (+) b = (a + b) modulo 30 [Divide (a + b) by 30 and take the remainder]
G is a group of order 3.
phi is a homomorphism from Z30 onto G.

Let ker(phi) denote the kernel of phi.

[The first isomorphism theorem states that if phi is a homomorphism from a group G onto H, (note that it must be onto), then the quotient group G/ker(phi) is isomorphic to H.]

Here, phi is a homomorphism from Z30 onto G. Therefore, by the first isomorphism theorem:
Z30/ker(phi) is isomorphic to G.
But G is of order three.
Hence Z30/ker(phi) is also of order 3, that is, it consists of 3 cosets.
Then, by Lagrange's theorem:
ker(phi) is of order 10.
Since Z30 is a cyclic group, and ker(phi) is a subgroup of Z30 of order 10 (which is a ...

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