Classify all groups of order 2012 up to isomorphism.
Hint: 503 is prime.
Solution
1. Let \(G\) be a group of order\[ |G|=2012=4\cdot 503. \]Its Sylow \(503\)-subgroup \(N\) is unique, hence normal, and \(N\cong C_{503}\).
The number \(n_{503}\) of Sylow \(503\)-subgroups satisfies \[ n_{503}\mid 4, \qquad n_{503}\equiv 1\pmod{503}. \] Thus \(n_{503}=1\). Since \(503\) is prime, \(N\cong C_{503}\).
2. If \(P\) is any Sylow \(2\)-subgroup of \(G\), then\[ G=N\rtimes P, \qquad |P|=4. \]Hence \(P\cong C_4\) or \(C_2\times C_2\).
Because \(N\trianglelefteq G\), the product \(NP\) is a subgroup. Also \[ N\cap P=1 \] because the two groups have coprime orders. Therefore \[ |NP|=|N||P|=503\cdot 4=|G|, \] so \(G=NP\), and this is the semidirect product \(N\rtimes P\). The two groups of order \(4\) are \(C_4\) and \(C_2\times C_2\).
3. The action of \(P\) on \(N\cong C_{503}\) has image of order at most \(2\).
Conjugation gives a homomorphism \[ \varphi:P\longrightarrow \operatorname{Aut}(N). \] Since \(N\cong C_{503}\), \[ \operatorname{Aut}(N)\cong (\mathbb Z/503\mathbb Z)^\times, \] a cyclic group of order \(502=2\cdot 251\). The image of the \(2\)-group \(P\) is therefore a \(2\)-subgroup of a cyclic group whose \(2\)-part has order \(2\). Hence \[ |\operatorname{im}\varphi|\in\{1,2\}. \] The unique automorphism of \(N\) of order \(2\) is inversion \(x\mapsto x^{-1}\).
4. If \(P\cong C_4\), there are exactly two semidirect products up to isomorphism:\[ C_{503}\times C_4 \quad\text{and}\quad C_{503}\rtimes C_4, \]where a generator of \(C_4\) acts on \(C_{503}\) by inversion.
By 3, the action is either trivial or has image the unique subgroup of order \(2\) in \(\operatorname{Aut}(C_{503})\). Since \(C_4\) has a unique quotient of order \(2\), there is only one nontrivial homomorphism \(C_4\to C_2\) up to automorphism of \(C_4\). Thus there is one trivial-action and one nontrivial-action isomorphism type.
5. If \(P\cong C_2\times C_2\), there are exactly two semidirect products up to isomorphism:\[ C_{503}\times C_2\times C_2 \quad\text{and}\quad C_{503}\rtimes (C_2\times C_2), \]where one quotient \(C_2\) of \(C_2\times C_2\) acts by inversion and the kernel of that quotient acts trivially.
Again the action is either trivial or has image \(C_2\). Any nontrivial homomorphism \[ C_2\times C_2\longrightarrow C_2 \] has kernel one of the three subgroups of order \(2\). The automorphism group of \(C_2\times C_2\) acts transitively on these three subgroups, so all nontrivial actions yield isomorphic semidirect products. Hence there is exactly one nontrivial-action type.
6. Consequently there are exactly four groups of order \(2012\), up to isomorphism:\[ \boxed{ C_{503}\times C_4, \quad C_{503}\times C_2\times C_2, \quad C_{503}\rtimes_{\mathrm{inv}} C_4, \quad C_{503}\rtimes_{\mathrm{inv}}(C_2\times C_2) }. \]
Steps <1>1–<1>5 show that every group of order \(2012\) is one of these four. They are pairwise nonisomorphic: the first two are abelian and have nonisomorphic Sylow \(2\)-subgroups; the last two are nonabelian and likewise have Sylow \(2\)-subgroups \(C_4\) and \(C_2\times C_2\), respectively.