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We describe abstract groups using standard building blocks:

• $C_n$ denotes the cyclic group of order $n$.
• $S_n$ denotes the symmetric group on $n$ letters.
• $A_n$ denotes the alternating group on $n$ letters.
• $D_n$ denotes the dihedral group of order $2n$.
• $Q_n$ denotes the quaternion group of order $n$.
• $\mathrm{GL}(n,q)$ denotes the general linear group of degree $n$ over the finite field of order $q$.
• $\mathrm{SL}(n,q)$ denotes the special linear group of degree $n$ over the finite field of order $q$.
• $SD_n$ denotes the semidihedral group or quasidihedral group of order $n=2^k$.
• $OD_n$ denotes the other dihedral group (or modular maximal-cyclic group) of order $n=2^k$. It is the non-trivial semidirect product $C_{2^{k-1}} : C_2$ which is not isomorphic to either $SD_n$ or $D_{2^{k-1}}$.
• $F_q$ denotes the Frobenius group for a prime power $q$. It is the group of affine linear transformations of the finite field $\mathbb{F}_q$. In other words, $F_q$ is a semidirect product $\mathbb{F}_q : \mathbb{F}_q^{\times}$.
• $He_p$ denotes the Heisenberg group, the unique non-abelian group of order $p^3$ and exponent $p$ for an odd prime $p$.

Groups $A$ and $B$ may be used to construct a larger group:

• $A\times B$ for the direct product of $A$ and $B$
• $A:B$ for the semidirect product of $A$ and $B$ (with normal subgroup $A$)
• $A.B$ an extension with normal subgroup $A$ and quotient isomorphic to $B$
• $A\wr B$ for the wreath product of $A$ and $B$
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Knowl status:
• Review status: reviewed
• Last edited by John Jones on 2022-06-29 13:11:40
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