# UCLA Smallest Integer Groups Regarding Embedding Degree an Abstract Algebra Worksheet

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The question is located in the image attached.Requirements: I would like each question to be rigorously proved with any theorems used to be cited to ensure that I can fully understand the question for my upcoming exam.

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1. Let G be a finite group. Let us say (only for this specific question) that the embedding
degree of G is the smallest positive integer n such that there exists an injective group
homomorphism 9:6 Sn from G to the symmetric group Sn. Equivalently, the
embedding degree of G is the smallest value of n such that G is isomorphic to a subgroup
of Sn. We write embd(G) for the embedding degree of G.
(a) Show that the embedding degree is equal to the smallest positive integer
n such that there exists a faithful action of G on a set with n elements.
(b) Show that, for any finite group G, we have embd(G) < Gl. Hint: remember Cayley's theorem, which we proved in lecture. (c) Let p be a prime number. Show that embd(Z/pZ) = p. Thus, the embedding degree of this particular group is as big as could possibly be allowed by part (b). Hint: show that if n Purchase answer to see full attachment Tags: homomorphism theorems embedding degree injectivity smallest integer User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

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